## Problem 522

Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is,
$H=\{ a\in G \mid \text{the order of a is finite}\}.$

Prove that $H$ is a subgroup of $G$. Add to solve later

## Problem 521

Decipher the trick of the following mathematical magic. Add to solve later

## Problem 520

Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$. Add to solve later

## Problem 519

Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD). Add to solve later

## Problem 518

Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). Add to solve later

## Problem 517

Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.
Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.

Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring. Add to solve later

## Problem 516

Prove the following statements.

(a) If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.

(b) Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$. Add to solve later

## Top 10 Popular Math Problems in 2016-2017

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?

The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level) Add to solve later

## Problem 514

Prove that a positive definite matrix has a unique positive definite square root. Add to solve later

## Problem 513

Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$.

Find all the square roots of the matrix
$A=\begin{bmatrix} 2 & 2\\ 2& 2 \end{bmatrix}.$ Add to solve later

## Problem 512

(a) Prove that the matrix $A=\begin{bmatrix} 0 & 1\\ 0& 0 \end{bmatrix}$ does not have a square root.
Namely, show that there is no complex matrix $B$ such that $B^2=A$.

(b) Prove that the $2\times 2$ identity matrix $I$ has infinitely many distinct square root matrices. Add to solve later

## Problem 511

Let $F$ be a finite field.
Prove that each element in the field $F$ is the sum of two squares in $F$. Add to solve later

## Problem 510

Let $(\Q, +)$ be the additive group of rational numbers and let $(\Q_{ > 0}, \times)$ be the multiplicative group of positive rational numbers.

Prove that $(\Q, +)$ and $(\Q_{ > 0}, \times)$ are not isomorphic as groups. Add to solve later

## Problem 509

Using the numbers appearing in
$\pi=3.1415926535897932384626433832795028841971693993751058209749\dots$ we construct the matrix $A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& 9 \end{bmatrix}.$

Prove that the matrix $A$ is nonsingular. Add to solve later

## Problem 508

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

## Problem 507

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

## 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
$(A^{\trans})^{-1}=(A^{-1})^{\trans}.$ Add to solve later

## 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 $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$. Add to solve later

## Problem 504

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

## 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. Add to solve later