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Abstract_algebra_0000 | Abstract_algebra | Fields and polynomials | Polynomials | 3 | [
"minimal polynomials"
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $5+2 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $5+2 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $5^{1/4}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}(c)$, where $c=\sqrt{15}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{15}$) | [
"x^2-10*x+29",
"x-(5+2*i)",
"x^4-5",
"x^4-16*x^2+4",
"x^2-2*c-8"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $-9+9 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $-9+9 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $2^{1/3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{13}+\sqrt{3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{13}+\sqrt{3}$ over $\mathbb{Q}(c)$, where $c=\sqrt{39}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{39}$) | [
"x^2+18*x+162",
"x-(9*i-9)",
"x^3-2",
"x^4-32*x^2+100",
"x^2-2*c-16"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $-4+2 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $-4+2 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $3^{1/3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}(c)$, where $c=\sqrt{15}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{15}$) | [
"x^2+8*x+20",
"x-(2*i-4)",
"x^3-3",
"x^4-16*x^2+4",
"x^2-2*c-8"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] |
Abstract_algebra_0001 | Abstract_algebra | Fields and polynomials | Polynomials | 4 | [
"quotient fields",
"polynomial rings"
] | Let $t \in \mathbb{Q}[x]/(x^2-11)$ be a root of the irreducible polynomial $x^2-11 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(6-t)(7+2t)$: [ANS] $+$ [ANS] $t$
(c) $(7+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(6-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"121",
"20",
"5",
"93",
"28",
"0.24",
"0.04"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | Let $t \in \mathbb{Q}[x]/(x^2-3)$ be a root of the irreducible polynomial $x^2-3 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(2-t)(1+2t)$: [ANS] $+$ [ANS] $t$
(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(2-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"9",
"-4",
"3",
"13",
"4",
"2",
"1"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | Let $t \in \mathbb{Q}[x]/(x^2-5)$ be a root of the irreducible polynomial $x^2-5 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(3-t)(1+2t)$: [ANS] $+$ [ANS] $t$
(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(3-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"25",
"-7",
"5",
"21",
"4",
"0.75",
"0.25"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0002 | Abstract_algebra | Fields and polynomials | Polynomials | [
"polynomials"
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{11}$ such that f(0)=6, \quad f(7)=7, \quad f(8)=8 $f(x)=$ [ANS] | [
"x^3+2*x^2+6"
] | [
"EX"
] | [
[]
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{3}$ such that f(0)=2, \quad f(1)=1, \quad f(2)=2 $f(x)=$ [ANS] | [
"x^3+x^2+2"
] | [
"EX"
] | [
[]
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{5}$ such that f(0)=3, \quad f(2)=2, \quad f(3)=3 $f(x)=$ [ANS] | [
"x^3+3*x^2+2*x+3"
] | [
"EX"
] | [
[]
] |
|
Abstract_algebra_0003 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"center of groups"
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"(e, x3)"
] | [
"UOL"
] | [
[]
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"e"
] | [
"EX"
] | [
[]
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0004 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"order of elements"
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x7)^2 & [ANS] \\ \hline (x8)^3 & [ANS] \\ \hline (x7)(x8) & [ANS] \\ \hline (x8)(x7) & [ANS] \\ \hline (x7)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x8$. Enter N if no such element exists. [ANS] | [
"x3",
"x11",
"x2",
"x4",
"x10",
"(e, x2, x4)",
"N",
"x11"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"EX"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x11)^2 & [ANS] \\ \hline (x2)^3 & [ANS] \\ \hline (x11)(x2) & [ANS] \\ \hline (x2)(x11) & [ANS] \\ \hline (x11)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x11$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x2$. Enter N if no such element exists. [ANS] | [
"x4",
"x2",
"x8",
"x10",
"x4",
"(e, x4, x5, x6, x7, x8, x9, x10, x11)",
"x4",
"x2"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"EX"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x7)^2 & [ANS] \\ \hline (x3)^3 & [ANS] \\ \hline (x7)(x3) & [ANS] \\ \hline (x3)(x7) & [ANS] \\ \hline (x7)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x3$. Enter N if no such element exists. [ANS] | [
"x2",
"x3",
"x10",
"x10",
"x11",
"(e, x2, x4)",
"N",
"(x1, x3, x5)"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"UOL"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0005 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"commutativity"
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS] | [
"(e, x3, x7, x10)"
] | [
"UOL"
] | [
[]
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x11$ and enter them as a comma-separated list. [ANS] | [
"(e, x4, x11)"
] | [
"UOL"
] | [
[]
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS] | [
"(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0006 | Abstract_algebra | Groups | Subgroups | 3 | [
"subgroups"
] | Find all elements of the subgroup $\langle 12 \rangle$ in $\mathbb{Z}_{84}$. [ANS]
Find all elements of the subgroup $\langle 9 \rangle$ in $\mathbb{Z}_{72}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 12, 24, 36, 48, 60, 72)",
"(0, 9, 18, 27, 36, 45, 54, 63)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements of the subgroup $\langle 6 \rangle$ in $\mathbb{Z}_{42}$. [ANS]
Find all elements of the subgroup $\langle 12 \rangle$ in $\mathbb{Z}_{96}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 6, 12, 18, 24, 30, 36)",
"(0, 12, 24, 36, 48, 60, 72, 84)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements of the subgroup $\langle 8 \rangle$ in $\mathbb{Z}_{56}$. [ANS]
Find all elements of the subgroup $\langle 11 \rangle$ in $\mathbb{Z}_{77}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 8, 16, 24, 32, 40, 48)",
"(0, 11, 22, 33, 44, 55, 66)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0007 | Abstract_algebra | Groups | Subgroups | 3 | [
"group tables",
"order of elements"
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\ \hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\ \hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\ \hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\ \hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\ \hline x_{6} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} \\ \hline x_{7} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{4} & x_{3} & x_{2} & x_{1} & e & x_{5} \\ \hline x_{8} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} & e \\ \hline x_{9} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & e & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} \\ \hline x_{10} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{1} & e & x_{5} & x_{4} & x_{3} & x_{2} \\ \hline x_{11} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{3} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{7} & [ANS] \\ \hline x_{8} & [ANS] \\ \hline x_{7} x_{8} & [ANS] \\ \hline (x_{7})^{2}(x_{8}) & [ANS] \\ \hline (x_{7})^{-1}(x_{8}) & [ANS] \\ \hline \end{array}$ | [
"4",
"4",
"3",
"4",
"6"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & e & x_{3} & x_{2} & x_{6} & x_{7} & x_{4} & x_{5} & x_{11} & x_{10} & x_{9} & x_{8} \\ \hline x_{2} & x_{2} & x_{3} & e & x_{1} & x_{7} & x_{6} & x_{5} & x_{4} & x_{9} & x_{8} & x_{11} & x_{10} \\ \hline x_{3} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{7} & x_{6} & x_{10} & x_{11} & x_{8} & x_{9} \\ \hline x_{4} & x_{4} & x_{7} & x_{5} & x_{6} & x_{11} & x_{8} & x_{10} & x_{9} & x_{2} & x_{1} & x_{3} & e \\ \hline x_{5} & x_{5} & x_{6} & x_{4} & x_{7} & x_{9} & x_{10} & x_{8} & x_{11} & x_{1} & x_{2} & e & x_{3} \\ \hline x_{6} & x_{6} & x_{5} & x_{7} & x_{4} & x_{8} & x_{11} & x_{9} & x_{10} & x_{3} & e & x_{2} & x_{1} \\ \hline x_{7} & x_{7} & x_{4} & x_{6} & x_{5} & x_{10} & x_{9} & x_{11} & x_{8} & e & x_{3} & x_{1} & x_{2} \\ \hline x_{8} & x_{8} & x_{10} & x_{11} & x_{9} & x_{1} & x_{3} & x_{2} & e & x_{7} & x_{5} & x_{4} & x_{6} \\ \hline x_{9} & x_{9} & x_{11} & x_{10} & x_{8} & x_{3} & x_{1} & e & x_{2} & x_{4} & x_{6} & x_{7} & x_{5} \\ \hline x_{10} & x_{10} & x_{8} & x_{9} & x_{11} & x_{2} & e & x_{1} & x_{3} & x_{6} & x_{4} & x_{5} & x_{7} \\ \hline x_{11} & x_{11} & x_{9} & x_{8} & x_{10} & e & x_{2} & x_{3} & x_{1} & x_{5} & x_{7} & x_{6} & x_{4} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{11} & [ANS] \\ \hline x_{2} & [ANS] \\ \hline x_{11} x_{2} & [ANS] \\ \hline (x_{11})^{2}(x_{2}) & [ANS] \\ \hline (x_{11})^{-1}(x_{2}) & [ANS] \\ \hline \end{array}$ | [
"3",
"2",
"3",
"3",
"3"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\ \hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\ \hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\ \hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\ \hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\ \hline x_{6} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ \hline x_{7} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e \\ \hline x_{8} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} \\ \hline x_{9} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} \\ \hline x_{10} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} \\ \hline x_{11} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{7} & [ANS] \\ \hline x_{3} & [ANS] \\ \hline x_{7} x_{3} & [ANS] \\ \hline (x_{7})^{2}(x_{3}) & [ANS] \\ \hline (x_{7})^{-1}(x_{3}) & [ANS] \\ \hline \end{array}$ | [
"6",
"2",
"6",
"6",
"6"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] |
Abstract_algebra_0008 | Abstract_algebra | Groups | Subgroups | 3 | [
"subgroups",
"generators"
] | Find all elements $x$ in $U(198)$ such that $\langle x \rangle$=$\langle 17 \rangle$: [ANS]
Also find all elements $x$ in $U(189)$ such that $\langle x \rangle$=$\langle 4 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(17, 161, 107, 35)",
"(4, 16, 67, 79, 130, 142)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements $x$ in $U(164)$ such that $\langle x \rangle$=$\langle 3 \rangle$: [ANS]
Also find all elements $x$ in $U(205)$ such that $\langle x \rangle$=$\langle 3 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(3, 27, 79, 55)",
"(3, 27, 38, 137)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements $x$ in $U(175)$ such that $\langle x \rangle$=$\langle 6 \rangle$: [ANS]
Also find all elements $x$ in $U(190)$ such that $\langle x \rangle$=$\langle 61 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(6, 41, 111, 146)",
"(61, 111, 161, 131, 101, 81)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0009 | Abstract_algebra | Groups | Subgroups | 4 | [
"subgroups",
"generators"
] | Find one pair of elements $x,$ $y$ in $U(14)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(14)$ and that $\langle x,y \rangle=U(14)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"9",
"13"
] | [
"NV",
"NV"
] | [
[]
] | Find one pair of elements $x,$ $y$ in $U(7)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(7)$ and that $\langle x,y \rangle=U(7)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"(2, 6)"
] | [
"OL"
] | [
[]
] | Find one pair of elements $x,$ $y$ in $U(8)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(8)$ and that $\langle x,y \rangle=U(8)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"(3, 5)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0010 | Abstract_algebra | Groups | Cyclic groups | 3 | [
"cyclic groups",
"generators"
] | Determine all generators of $\mathbb{Z}_{21}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20)"
] | [
"UOL"
] | [
[]
] | Determine all generators of $\mathbb{Z}_{15}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 4, 7, 8, 11, 13, 14)"
] | [
"UOL"
] | [
[]
] | Determine all generators of $\mathbb{Z}_{18}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 7, 11, 13, 17)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0011 | Abstract_algebra | Groups | Cyclic groups | 3 | [
"cyclic groups",
"order of elements"
] | (a) Find all elements in $\{ cyclic(143) \}$ of order $13$. [ANS]
(b) Find all elements in the subgroup $\langle 13 \rangle$ of $\{ cyclic(143) \}$. [ANS] | [
"(11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132)",
"(0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all elements in $\{ cyclic(51) \}$ of order $3$. [ANS]
(b) Find all elements in the subgroup $\langle 3 \rangle$ of $\{ cyclic(51) \}$. [ANS] | [
"(17, 34)",
"(0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all elements in $\{ cyclic(55) \}$ of order $5$. [ANS]
(b) Find all elements in the subgroup $\langle 5 \rangle$ of $\{ cyclic(55) \}$. [ANS] | [
"(11, 22, 33, 44)",
"(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0012 | Abstract_algebra | Groups | Cyclic groups | 4 | [
"cyclic groups",
"generators",
"subgroups"
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{35}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{35}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 5, 7, 10, 14, 15, 20, 21, 25, 28, 30)"
] | [
"UOL"
] | [
[]
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{33}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{33}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 3, 6, 9, 11, 12, 15, 18, 21, 22, 24, 27, 30)"
] | [
"UOL"
] | [
[]
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{28}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{28}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0013 | Abstract_algebra | Groups | Cyclic groups | 6 | [
"cyclic groups",
"generators",
"subgroups"
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{1309}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{1309}$, and that $\langle x, y \rangle=\mathbb{Z}_{1309}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 1309 is a product of three primes. | [
"(17, 11)"
] | [
"UOL"
] | [
[]
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{114}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{114}$, and that $\langle x, y \rangle=\mathbb{Z}_{114}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 114 is a product of three primes. | [
"(2, 19)"
] | [
"UOL"
] | [
[]
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{195}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{195}$, and that $\langle x, y \rangle=\mathbb{Z}_{195}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 195 is a product of three primes. | [
"(5, 13)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0014 | Abstract_algebra | Groups | Cyclic groups | 4 | [
"cyclic groups",
"order of groups",
"order of elements",
"subgroups"
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=18$ and $\textrm{ord}(y)=24$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 2, 3, 6)"
] | [
"OL"
] | [
[]
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=12$ and $\textrm{ord}(y)=15$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 3)"
] | [
"OL"
] | [
[]
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=14$ and $\textrm{ord}(y)=21$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 7)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0015 | Abstract_algebra | Groups | Cyclic groups | 2 | [
"cyclic groups",
"order of elements"
] | Determine the order of every element of $\mathbb{Z}_{21}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 21, 21, 7, 21, 21, 7, 3, 21, 7, 21, 21, 7, 21, 3, 7, 21, 21, 7, 21, 21)"
] | [
"OL"
] | [
[]
] | Determine the order of every element of $\mathbb{Z}_{15}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 15, 15, 5, 15, 3, 5, 15, 15, 5, 3, 15, 5, 15, 15)"
] | [
"OL"
] | [
[]
] | Determine the order of every element of $\mathbb{Z}_{18}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 18, 9, 6, 9, 18, 3, 18, 9, 2, 9, 18, 3, 18, 9, 6, 9, 18)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0017 | Abstract_algebra | Groups | Product of groups | 4 | [
"product of groups",
"generators"
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{176}$ such that $\mathbb{Z}_{176}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(16, 11)"
] | [
"UOL"
] | [
[]
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{75}$ such that $\mathbb{Z}_{75}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(3, 25)"
] | [
"UOL"
] | [
[]
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{77}$ such that $\mathbb{Z}_{77}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(7, 11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0018 | Abstract_algebra | Groups | Product of groups | 4 | [
"products of groups",
"generators"
] | Find a pair of elements $A, B$ in $U(63)$ such that $U(63)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 63$. $A, B=$ [ANS] | [
"(10, 29)"
] | [
"UOL"
] | [
[]
] | Find a pair of elements $A, B$ in $U(69)$ such that $U(69)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 69$. $A, B=$ [ANS] | [
"(47, 28)"
] | [
"UOL"
] | [
[]
] | Find a pair of elements $A, B$ in $U(91)$ such that $U(91)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 91$. $A, B=$ [ANS] | [
"(66, 15)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0019 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets"
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{80}$, $H=10 \mathbb{Z}_{80}$, $g=53 \pmod{80}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<k\right>$ $g$ $=k$ [ANS] | [
"(53, 63, 73, 3, 13, 23, 33, 43)",
"(1, -1, k, -k)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{56}$, $H=14 \mathbb{Z}_{56}$, $g=12 \pmod{56}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<j\right>$ $g$ $=-i$ [ANS] | [
"(12, 26, 40, 54)",
"(-i, i, k, -k)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{55}$, $H=11 \mathbb{Z}_{55}$, $g=19 \pmod{55}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<j\right>$ $g$ $=k$ [ANS] | [
"(19, 30, 41, 52, 8)",
"(k, -k, i, -i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0020 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets",
"coset representatives"
] | Find a complete set of coset representatives of the subgroup $\langle 19 \rangle$ in $U(56)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 56$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 8 \rangle$ in $U(51)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 51$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 5, 11, 29)",
"(1, 5, 11, 19)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find a complete set of coset representatives of the subgroup $\langle 4 \rangle$ in $U(35)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 35$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 20 \rangle$ in $U(63)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 63$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 2, 3, 6)",
"(1, 2, 4, 5, 8, 10)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find a complete set of coset representatives of the subgroup $\langle 37 \rangle$ in $U(42)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 42$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 21 \rangle$ in $U(52)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 52$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 5, 11, 13)",
"(1, 3, 7, 9, 27, 29)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0022 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 6 | [
"Lagrange theorem"
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=35$ and $\#G=350$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(70, 175)"
] | [
"UOL"
] | [
[]
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=22$ and $\#G=308$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(44, 154)"
] | [
"UOL"
] | [
[]
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=26$ and $\#G=260$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(52, 130)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0023 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets"
] | (a) Determine all elements of the coset $4+\langle 10 \rangle$ in the group $\mathbb{Z}_{14}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 14$. [ANS]
(b) Determine all elements of the coset $1 \langle 26 \rangle$ in the group $U(57)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 57$. [ANS] | [
"(4, 6, 8, 10, 12, 0, 2)",
"(26, 49, 20, 7, 11, 1)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Determine all elements of the coset $1+\langle 6 \rangle$ in the group $\mathbb{Z}_{6}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 6$. [ANS]
(b) Determine all elements of the coset $2 \langle 26 \rangle$ in the group $U(57)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 57$. [ANS] | [
"1",
"(52, 41, 40, 14, 22, 2)"
] | [
"NV",
"UOL"
] | [
[],
[]
] | (a) Determine all elements of the coset $2+\langle 6 \rangle$ in the group $\mathbb{Z}_{9}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 9$. [ANS]
(b) Determine all elements of the coset $1 \langle 41 \rangle$ in the group $U(72)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 72$. [ANS] | [
"(2, 5, 8)",
"(41, 25, 17, 49, 65, 1)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0024 | Abstract_algebra | Groups | Homomorphisms | 6 | [
"group homomorphisms",
"cyclic groups",
"order of groups",
"image"
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{245} \rightarrow \mathbb{Z}_{175}$
such that the image of $f$ has size exactly $7$.
First, since $245 \equiv 0$ $\pmod{245}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(245) \\ & \equiv & 245 f(1) \pmod{175} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{175}$ so $0 \equiv 175 f(1) \pmod{175}$.
Thus $\gcd(245,175) f(1) \equiv 0$ $\pmod{175}$ i.e. $35 f(1) \equiv 0$ $\pmod{175}$ whence $(\ast)$ $f(1)=5 u$ for some $u$ in $\mathbb{Z}_{175}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{175}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $7$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $7$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 25, 50, 75, 100, 125, 150)",
"(25, 50, 75, 100, 125, 150)",
"6"
] | [
"OL",
"OL",
"NV"
] | [
[],
[],
[]
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{28} \rightarrow \mathbb{Z}_{98}$
such that the image of $f$ has size exactly $2$.
First, since $28 \equiv 0$ $\pmod{28}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(28) \\ & \equiv & 28 f(1) \pmod{98} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{98}$ so $0 \equiv 98 f(1) \pmod{98}$.
Thus $\gcd(28,98) f(1) \equiv 0$ $\pmod{98}$ i.e. $14 f(1) \equiv 0$ $\pmod{98}$ whence $(\ast)$ $f(1)=7 u$ for some $u$ in $\mathbb{Z}_{98}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{98}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $2$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $2$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 49)",
"49",
"1"
] | [
"OL",
"NV",
"NV"
] | [
[],
[],
[]
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{45} \rightarrow \mathbb{Z}_{75}$
such that the image of $f$ has size exactly $3$.
First, since $45 \equiv 0$ $\pmod{45}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(45) \\ & \equiv & 45 f(1) \pmod{75} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{75}$ so $0 \equiv 75 f(1) \pmod{75}$.
Thus $\gcd(45,75) f(1) \equiv 0$ $\pmod{75}$ i.e. $15 f(1) \equiv 0$ $\pmod{75}$ whence $(\ast)$ $f(1)=5 u$ for some $u$ in $\mathbb{Z}_{75}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{75}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $3$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $3$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 25, 50)",
"(25, 50)",
"2"
] | [
"OL",
"OL",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0025 | Abstract_algebra | Groups | Group actions | 6 | [
"group actions",
"orbit-stabilizer theorem"
] | Let $G$ be a finite group of order $77$ acting on a finite set $S$ of size $11$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 11)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $7$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 3, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $15$ acting on a finite set $S$ of size $5$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 3, 5)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0026 | Abstract_algebra | Groups | Group actions | 6 | [
"group actions",
"orbit-stabilizer theorem"
] | Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $34$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $49$ acting on a finite set $S$ of size $17$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $36$ acting on a finite set $S$ of size $23$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 3, 4, 6, 9, 12, 18)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0027 | Abstract_algebra | Groups | Group actions | 4 | [
"group actions",
"conjugation",
"conjugacy classes"
] | In this problem we determine the conjugacy class of the elements $a^{6} b^0$ and $a^{6} b^1$ in the dihedral group $D_{11}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{6} b^0? & is a^i b^1conjugate to a^{6} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 7 & [ANS] & [ANS] \\ \hline 8 & [ANS] & [ANS] \\ \hline 9 & [ANS] & [ANS] \\ \hline 10 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"Y",
"Y",
"Y",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{6}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"N",
"Y",
"Y",
"N",
"N",
"N",
"Y",
"N",
"N",
"Y",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{8}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 7 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"N",
"N",
"Y",
"N",
"N",
"Y",
"Y",
"N",
"N",
"Y",
"Y",
"N",
"N",
"N",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0028 | Abstract_algebra | Rings | Ring axioms | 2 | [
"ring axioms"
] | Let $a=148$ and $b=157$ be elements in the ring $\mathbb{Z}_{175}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 175$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"130",
"166",
"136"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Let $a=40$ and $b=62$ be elements in the ring $\mathbb{Z}_{63}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 63$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"39",
"41",
"23"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Let $a=56$ and $b=60$ be elements in the ring $\mathbb{Z}_{75}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 75$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"41",
"71",
"60"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0029 | Abstract_algebra | Rings | Ring axioms | 3 | [
"ring axioms"
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6,7,8,9} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {1,2,4,5,8,9} \rbrace
B=\lbrace {1,2,7,9} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"512",
"(1, 2, 4, 5, 8, 9)",
"(4, 5, 7, 8)",
"(1, 2, 9)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {2,3,4,5,6} \rbrace
B=\lbrace {3,4,5} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"64",
"(2, 3, 4, 5, 6)",
"(2, 6)",
"(3, 4, 5)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6,7} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {1,3,4,6,7} \rbrace
B=\lbrace {1,2,4} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"128",
"(1, 3, 4, 6, 7)",
"(2, 3, 6, 7)",
"(1, 4)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] |
Abstract_algebra_0030 | Abstract_algebra | Rings | Ring axioms | 3 | [
"ring axioms",
"inverse"
] | (a) Find the multiplicative inverse of $38$ in $\mathbb{Z}_{39}$. [ANS]
(b) Find the multiplicative inverse of $30$ in $\mathbb{Z}_{31}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"38",
"30",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] | (a) Find the multiplicative inverse of $8$ in $\mathbb{Z}_{9}$. [ANS]
(b) Find the multiplicative inverse of $46$ in $\mathbb{Z}_{47}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"8",
"46",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] | (a) Find the multiplicative inverse of $19$ in $\mathbb{Z}_{20}$. [ANS]
(b) Find the multiplicative inverse of $32$ in $\mathbb{Z}_{33}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"19",
"32",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] |
Abstract_algebra_0031 | Abstract_algebra | Rings | Units and zero divisors | 4 | [
"commutativity",
"zero-divisors"
] | Denote by $R$ the set of all functions from the set $\lbrace-8, 8, 9, 10 \rbrace$ to the ring $\mathbb{Z}_{45}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-8 & [ANS] \\ \hline 8 & [ANS] \\ \hline 9 & [ANS] \\ \hline 10 & [ANS] \\ \hline \end{array}$ | [
"Y",
"331776",
"(3, 0, 0, 0)"
] | [
"TF",
"NV",
"UOL"
] | [
[],
[],
[]
] | Denote by $R$ the set of all functions from the set $\lbrace-10, 2, 8, 10 \rbrace$ to the ring $\mathbb{Z}_{30}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-10 & [ANS] \\ \hline 2 & [ANS] \\ \hline 8 & [ANS] \\ \hline 10 & [ANS] \\ \hline \end{array}$ | [
"Y",
"4096",
"(2, 0, 0, 0)"
] | [
"TF",
"NV",
"UOL"
] | [
[],
[],
[]
] | Denote by $R$ the set of all functions from the set $\lbrace-8,-1, 2, 8 \rbrace$ to the ring $\mathbb{Z}_{18}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-8 & [ANS] \\ \hline-1 & [ANS] \\ \hline 2 & [ANS] \\ \hline 8 & [ANS] \\ \hline \end{array}$ | [
"TF",
"1296",
"(2, 0, 0, 0)"
] | [
"EX",
"NV",
"UOL"
] | [
[],
[],
[]
] |
Abstract_algebra_0032 | Abstract_algebra | Rings | Units and zero divisors | 3 | [
"characteristic"
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{46} \times \mathbb{Z}_{36}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{38} \times \mathbb{Z}_{21}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{35}$. [ANS] | [
"828",
"798",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{6} \times \mathbb{Z}_{56}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{10} \times \mathbb{Z}_{21}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{57}$. [ANS] | [
"168",
"210",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{20} \times \mathbb{Z}_{18}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{34} \times \mathbb{Z}_{49}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{55}$. [ANS] | [
"180",
"1666",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0033 | Abstract_algebra | Rings | Units and zero divisors | 4 | [
"characteristic",
"integral domains"
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{36} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{44} \times \mathbb{Z}_{19} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{41} \times \mathbb{Z}_{41} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{38}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"836",
"N",
"41",
"N",
"0",
"Y",
"38",
"N"
] | [
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{56} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{21} \times \mathbb{Z}_{57} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{3} \times \mathbb{Z}_{3} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{10}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"399",
"N",
"3",
"N",
"0",
"Y",
"10",
"N"
] | [
"NV",
"EX",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{37} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{34} \times \mathbb{Z}_{14} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{13} \times \mathbb{Z}_{13} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{18}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"238",
"N",
"13",
"N",
"0",
"Y",
"18",
"N"
] | [
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0034 | Abstract_algebra | Rings | Units and zero divisors | 2 | [
"inverse",
"zero-divisors"
] | For each of the following elements of the ring $\mathbb{Z}_{45}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 25 & [ANS] & [ANS] \\ \hline 31 & [ANS] & [ANS] \\ \hline 14 & [ANS] & [ANS] \\ \hline 15 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"Y",
"0",
"N",
"16",
"N",
"29",
"Y",
"0"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following elements of the ring $\mathbb{Z}_{30}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 9 & [ANS] & [ANS] \\ \hline 13 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 23 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"Y",
"0",
"N",
"7",
"Y",
"0",
"N",
"17"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following elements of the ring $\mathbb{Z}_{18}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 5 & [ANS] & [ANS] \\ \hline 11 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 15 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"11",
"N",
"5",
"Y",
"0",
"Y",
"0"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0035 | Abstract_algebra | Rings | Units and zero divisors | 3 | [
"zero-divisors"
] | For which of the integers $n \in \lbrace 34, 59, 73, 25, 30, 50, 58, 38 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $45x=45 y$ then $x=y$? [ANS] | [
"(73, 34, 58, 59, 38)"
] | [
"UOL"
] | [
[]
] | For which of the integers $n \in \lbrace 37, 35, 34, 19, 32, 18, 7, 95 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $30x=30 y$ then $x=y$? [ANS] | [
"(7, 19, 37)"
] | [
"UOL"
] | [
[]
] | For which of the integers $n \in \lbrace 87, 92, 35, 82, 21, 29, 55, 19 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $18x=18 y$ then $x=y$? [ANS] | [
"(55, 35, 19, 29)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0036 | Abstract_algebra | Rings | Units and zero divisors | 6 | [
"zero divisors",
"Chinese remainder theorem"
] | Find all integers $n$ in the set $\lbrace 33, 31, 48, 29, 24, 23, 26, 53, 42 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(31, 53, 23, 29)"
] | [
"UOL"
] | [
[]
] | Find all integers $n$ in the set $\lbrace 26, 38, 41, 43, 48, 31, 42 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(41, 31, 43)"
] | [
"UOL"
] | [
[]
] | Find all integers $n$ in the set $\lbrace 37, 42, 26, 24, 38, 23, 53, 28, 43, 47, 33 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(37, 47, 43, 23, 53)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0037 | Abstract_algebra | Rings | Ideals and homomorphisms | 6 | [
"ring homomorphisms"
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{45} \rightarrow \mathbb{Z}_{45}$: [ANS]
(b) $\mathbb{Z}_{33} \rightarrow \mathbb{Z}_{11}$: [ANS]
(c) $\mathbb{Z}_{93} \rightarrow \mathbb{Z}_{16}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"45",
"11",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{23} \rightarrow \mathbb{Z}_{23}$: [ANS]
(b) $\mathbb{Z}_{55} \rightarrow \mathbb{Z}_{8}$: [ANS]
(c) $\mathbb{Z}_{72} \rightarrow \mathbb{Z}_{24}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"23",
"0",
"24"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{35} \rightarrow \mathbb{Z}_{35}$: [ANS]
(b) $\mathbb{Z}_{139} \rightarrow \mathbb{Z}_{21}$: [ANS]
(c) $\mathbb{Z}_{36} \rightarrow \mathbb{Z}_{9}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"35",
"0",
"9"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0038 | Abstract_algebra | Rings | Ideals and homomorphisms | 2 | [
"ideals"
] | In the ring $\mathbb{Z}_{88}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 88$.
(a) $(51)+(55)$ $($ [ANS] $)$
(b) $(51)(55)$ $($ [ANS] $)$
(c) $(51) \cap (55)$ $($ [ANS] $)$ | [
"1",
"77",
"77"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | In the ring $\mathbb{Z}_{54}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 54$.
(a) $(49)+(9)$ $($ [ANS] $)$
(b) $(49)(9)$ $($ [ANS] $)$
(c) $(49) \cap (9)$ $($ [ANS] $)$ | [
"1",
"9",
"9"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | In the ring $\mathbb{Z}_{65}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 65$.
(a) $(39)+(18)$ $($ [ANS] $)$
(b) $(39)(18)$ $($ [ANS] $)$
(c) $(39) \cap (18)$ $($ [ANS] $)$ | [
"3",
"52",
"52"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0039 | Abstract_algebra | Rings | Ideals and homomorphisms | 3 | [
"maximal ideals"
] | It is a fact that every ideal of $\mathbb{Z}_{144}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{144}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{144}$ containing the ideal $(64)$. Enter a generator for each of these ideals. That is, if you think $(64)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"2"
] | [
"NV"
] | [
[]
] | It is a fact that every ideal of $\mathbb{Z}_{42}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{42}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{42}$ containing the ideal $(36)$. Enter a generator for each of these ideals. That is, if you think $(36)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"(2, 3)"
] | [
"UOL"
] | [
[]
] | It is a fact that every ideal of $\mathbb{Z}_{70}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{70}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{70}$ containing the ideal $(50)$. Enter a generator for each of these ideals. That is, if you think $(50)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"(2, 5)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0040 | Abstract_algebra | Rings | Ideals and homomorphisms | 3 | [
"ideals",
"generators"
] | It is a fact that every ideal of $\mathbb{Z}_{72}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{72}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{72}$ that are contained in the ideal $(162)$: $(162) \subseteq I \subseteq \mathbb{Z}_{72}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{72}$ that contain the ideal $(162)$: J \subseteq (162) \subseteq \mathbb{Z}_{72}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 18, 36, 54)",
"(1, 2, 3, 6, 9, 18)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | It is a fact that every ideal of $\mathbb{Z}_{40}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{40}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{40}$ that are contained in the ideal $(56)$: $(56) \subseteq I \subseteq \mathbb{Z}_{40}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{40}$ that contain the ideal $(56)$: J \subseteq (56) \subseteq \mathbb{Z}_{40}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 8, 16, 24, 32)",
"(1, 2, 4, 8)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | It is a fact that every ideal of $\mathbb{Z}_{108}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{108}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{108}$ that are contained in the ideal $(60)$: $(60) \subseteq I \subseteq \mathbb{Z}_{108}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{108}$ that contain the ideal $(60)$: J \subseteq (60) \subseteq \mathbb{Z}_{108}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 12, 24, 36, 48, 60, 72, 84, 96)",
"(1, 2, 3, 4, 6, 12)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0041 | Abstract_algebra | Rings | Ideals and homomorphisms | 2 | [
"ideals"
] | (a) Determine all elements in the ideal $(10)$ of $\mathbb{Z}_{30}$. [ANS]
(b) Determine all elements in the ideal $(10)+(12)$ of $\mathbb{Z}_{30}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{30}$ such that $(10)+(m)$ is a proper ideal of $\mathbb{Z}_{30}$. [ANS] | [
"(0, 10, 20)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28)",
"(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] | (a) Determine all elements in the ideal $(14)$ of $\mathbb{Z}_{28}$. [ANS]
(b) Determine all elements in the ideal $(14)+(12)$ of $\mathbb{Z}_{28}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{28}$ such that $(14)+(m)$ is a proper ideal of $\mathbb{Z}_{28}$. [ANS] | [
"(0, 14)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26)",
"(0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] | (a) Determine all elements in the ideal $(10)$ of $\mathbb{Z}_{20}$. [ANS]
(b) Determine all elements in the ideal $(10)+(8)$ of $\mathbb{Z}_{20}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{20}$ such that $(10)+(m)$ is a proper ideal of $\mathbb{Z}_{20}$. [ANS] | [
"(0, 10)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18)",
"(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] |
Abstract_algebra_0042 | Abstract_algebra | Rings | Quotient rings and polynomial rings | 5 | [
"quotient rings",
"polynomial rings"
] | Find all elements $b \in \mathbb{Z}_{7}$ such that the quotient ring
$\mathbb{Z}_{7} [x]/(x^2+4*x+b)$ is a field. [ANS] | [
"(1, 5, 6)"
] | [
"UOL"
] | [
[]
] | Find all elements $b \in \mathbb{Z}_{2}$ such that the quotient ring
$\mathbb{Z}_{2} [x]/(x^2+x+b)$ is a field. [ANS] | [
"1"
] | [
"NV"
] | [
[]
] | Find all elements $b \in \mathbb{Z}_{3}$ such that the quotient ring
$\mathbb{Z}_{3} [x]/(x^2+x+b)$ is a field. [ANS] | [
"2"
] | [
"NV"
] | [
[]
] |
Abstract_algebra_0043 | Abstract_algebra | Rings | Quotient rings and polynomial rings | 3 | [
"polynomials rings",
"associates"
] | (a) Find all associates of $(7*x^4+7*x^3+8)$ in $\mathbb{Z}_{12} [x]$. Make sure the coefficients are $\geq 0$ and $< 12$. [ANS]
(b) Find all associates of $(1+i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(7*x^4+7*x^3+8, 11*x^4+11*x^3+4, x^4+x^3+8, 5*x^4+5*x^3+4)",
"(1+i, -1-i, -1+i, 1-i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all associates of $(5*x^9+x^3+2)$ in $\mathbb{Z}_{6} [x]$. Make sure the coefficients are $\geq 0$ and $< 6$. [ANS]
(b) Find all associates of $(-6-3i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(5*x^9+x^3+2, x^9+5*x^3+4)",
"(-6-3i, 6+3i, 3-6i, -3+6i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all associates of $(5*x^9+3*x^7+6)$ in $\mathbb{Z}_{8} [x]$. Make sure the coefficients are $\geq 0$ and $< 8$. [ANS]
(b) Find all associates of $(-6-4i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(5*x^9+3*x^7+6, 7*x^9+x^7+2, x^9+7*x^7+6, 3*x^9+5*x^7+2)",
"(-6-4i, 6+4i, 4-6i, -4+6i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
UGMathBench: A Diverse and Dynamic Benchmark for Undergraduate-Level Mathematical Reasoning with Large Language Models
UGMathBench is a diverse and dynamic benchmark specifically designed for evaluating undergraduate-level mathematical reasoning with LLMs. UGMathBench comprises 5,062 problems across 16 subjects and 111 topics, featuring 10 distinct answer types. Each problem includes three randomized versions.
An Example to load the data
from datasets import load_dataset
dataset=load_dataset("UGMathBench/ugmathbench", "Trigonometry", split="test")
print(dataset[0])
More details on loading and using the data are on our GitHub page.
If you do find our code helpful or use our benchmark dataset, please cite our paper.
@article{xu2025ugmathbench,
title={UGMathBench: A Diverse and Dynamic Benchmark for Undergraduate-Level Mathematical Reasoning with Large Language Models},
author={Xu, Xin and Zhang, Jiaxin and Chen, Tianhao and Chao, Zitong and Hu, Jishan and Yang, Can},
journal={arXiv preprint arXiv:2501.13766},
year={2025}
}
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