## Problem 508

Let $A$ be a square matrix.
Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

## Problem 507

Prove that any field automorphism of the field of real numbers $\R$ must be the identity automorphism.

## Problem 506

Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.
Namely, show that
$(A^{\trans})^{-1}=(A^{-1})^{\trans}.$

## Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
$(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$

Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.

## Problem 504

Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.

## Problem 503

Prove that the ring of integers
$\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}$ of the field $\Q(\sqrt{2})$ is a Euclidean Domain.

## Problem 502

Find the inverse matrix of the $3\times 3$ matrix
$A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}$ using the Cayley-Hamilton theorem.

## 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.

## Problem 500

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.

## Problem 499

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 $\C$.

## Problem 498

Let $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is 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
$\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}.$

## Problem 497

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.

## Problem 496

Prove 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 $2^n-1$.

## Problem 495

Prove that every finite group having more than two elements has a nontrivial automorphism.

(Michigan State University, Abstract Algebra Qualifying Exam)

## Problem 494

Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.

## Problem 493

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

## Problem 492

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.

## Problem 491

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 $\Q(i, \sqrt{2})$.

## Problem 490

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.

## Problem 489

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