## Every Ring of Order $p^2$ is Commutative

## Problem 501

Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.

Then prove that $R$ is a commutative ring.

of the day

Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.

Then prove that $R$ is a commutative ring.

10 questions about nonsingular matrices, invertible matrices, and linearly independent vectors.

The quiz is designed to test your understanding of the basic properties of these topics.

You can take the quiz as many times as you like.

The solutions will be given after completing all the 10 problems.

Click the **View question** button to see the solutions.

Find an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.

Add to solve laterLet $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself of the reflection across a line $y=mx$ for some $m\in \R$.

Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}.\]

Let $G$ be an abelian group.

Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.

Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

Add to solve laterProve that if $2^n-1$ is a Mersenne prime number, then

\[N=2^{n-1}(2^n-1)\]
is a perfect number.

On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.

Add to solve laterProve that every finite group having more than two elements has a nontrivial automorphism.

(Michigan State University, Abstract Algebra Qualifying Exam)

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Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying

\[|A|+|B| > |G|.\]
Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.

Then prove that $G=AB$, where

\[AB=\{ab \mid a\in A, b\in B\}.\]

Let

\[D=\begin{bmatrix}

d_1 & 0 & \dots & 0 \\

0 &d_2 & \dots & 0 \\

\vdots & & \ddots & \vdots \\

0 & 0 & \dots & d_n

\end{bmatrix}\]
be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.

Let $A=(a_{ij})$ be an $n\times n$ matrix such that $A$ commutes with $D$, that is,

\[AD=DA.\]
Then prove that $A$ is a diagonal matrix.

Let $\zeta_8$ be a primitive $8$-th root of unity.

Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.

Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.

Define a map $\tilde{f}:H\to K$ as follows.

For each $h\in H$, there exists $g\in G$ such that $\pi(g)=h$ since $\pi:G\to H$ is surjective.

Define $\tilde{f}:H\to K$ by $\tilde{f}(h)=f(g)$.

**(a)** Prove that the map $\tilde{f}:H\to K$ is well-defined.

**(b)** Prove that $\tilde{f}:H\to K$ is a group homomorphism.

Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.

Prove that $\alpha$ is an integer.

Let $G$ be a finite group and let $S$ be a non-empty set.

Suppose that $G$ acts on $S$ freely and transitively.

Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.

Let

\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]
be an ideal of the ring

\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]
Then determine the quotient ring $\Z[\sqrt{10}]/P$.

Is $P$ a prime ideal? Is $P$ a maximal ideal?

Determine whether there exists a nonsingular matrix $A$ if

\[A^4=ABA^2+2A^3,\]
where $B$ is the following matrix.

\[B=\begin{bmatrix}

-1 & 1 & -1 \\

0 &-1 &0 \\

2 & 1 & -4

\end{bmatrix}.\]

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(The Ohio State University, Linear Algebra Final Exam Problem)

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Let

\[A=\begin{bmatrix}

1 & -14 & 4 \\

-1 &6 &-2 \\

-2 & 24 & -7

\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}

4 \\

-1 \\

-7

\end{bmatrix}.\]
Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.

The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by

\[E_{-1}=\Span\left\{\, \begin{bmatrix}

3 \\

-1 \\

-5

\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}

-2 \\

1 \\

4

\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}

-4 \\

2 \\

7

\end{bmatrix} \,\right\}.\]

(The Ohio State University, Linear Algebra Final Exam Problem)

Add to solve later Let $A$ be a square matrix and its characteristic polynomial is give by

\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.

(The Ohio State University, Linear Algebra Final Exam Problem)

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Diagonalize the matrix

\[A=\begin{bmatrix}

1 & 1 & 1 \\

1 &1 &1 \\

1 & 1 & 1

\end{bmatrix}.\]
Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

For which values of constants $a, b$ and $c$ is the matrix

\[A=\begin{bmatrix}

7 & a & b \\

0 &2 &c \\

0 & 0 & 3

\end{bmatrix}.\]
diagonalizable?

(The Ohio State University, Linear Algebra Final Exam Problem)

Add to solve later