## The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers

## Problem 534

Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

Add to solve laterLet $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

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 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.

**(a)** Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$.

**(b)** Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.

Consider the matrix

\[A=\begin{bmatrix}

3/2 & 2\\

-1& -3/2

\end{bmatrix} \in M_{2\times 2}(\R).\]

**(a)** Find the eigenvalues and corresponding eigenvectors of $A$.

**(b)** Show that for $\mathbf{v}=\begin{bmatrix}

1 \\

0

\end{bmatrix}\in \R^2$, we can choose $n$ large enough so that the length $\|A^n\mathbf{v}\|$ is as small as we like.

(*University of California, Berkeley, Linear Algebra Final Exam Problem*)

Read solution

Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$.

Prove the **Cauchy-Schwarz inequality**:

\[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]

In the ring

\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\]
show that $5$ is a prime element but $7$ is not a prime element.

Show that eigenvalues of a Hermitian matrix $A$ are real numbers.

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

Read solution

Denote by $i$ the square root of $-1$.

Let

\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]
be the ring of Gaussian integers.

We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to

\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]

Here $\bar{\alpha}$ is the complex conjugate of $\alpha$.

Then show that an element $\alpha \in R$ is a unit if and only if the norm $N(\alpha)=\pm 1$.

Also, determine all the units of the ring $R=\Z[i]$ of Gaussian integers.