## The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)

## Problem 519

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

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

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

Add to solve later 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 laterProve 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 $R$.

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)

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Prove that a positive definite matrix has a unique positive definite square root.

Add to solve later 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}.\]

**(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.

Let $F$ be a finite field.

Prove that each element in the field $F$ is the sum of two squares in $F$.

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 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 Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

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

Add to solve laterLet $A$ 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

\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]

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}$.

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

Add to solve later 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.

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.

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.

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