## $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.

## Problem 239

Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Add to solve laterLet $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Add to solve laterHappy New Year 2017!!

Here is the list of mathematical facts about **the number 2017** that you can brag about to your friends or family as a math geek.

Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$.

Then show that the determinant \[ \det(I-A) >0,\] where $I$ is the $n \times n$ identity matrix.

Add to solve laterLet $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer.

Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.

Add to solve laterSuppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$.

Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$.

(*Nagoya University, Linear Algebra Final Exam Problem*)

Read solution

Show that the polynomial

\[f(x)=x^4-2x-1\]
is irreducible over the field of rational numbers $\Q$.

**(a)** Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.

**(b)** Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.

**(c)** Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.

Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.

Add to solve laterShow that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group.

Add to solve laterLet $\Q$ be the field of rational numbers.

**(a)** Is the polynomial $f(x)=x^2-2$ separable over $\Q$?

**(b)** Find the Galois group of $f(x)$ over $\Q$.

Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements.

For any nonzero element $a\in \F_p$, prove that the polynomial

\[f(x)=x^p-x+a\]
is irreducible and separable over $F_p$.

(Dummit and Foote “Abstract Algebra” Section 13.5 Exercise #5 on p.551)

Add to solve laterLet $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.

Add to solve laterSuppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.

Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.

Then show that $N$ contains all $p$-Sylow subgroups of $G$.

Add to solve laterLet $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.

Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.

Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.

Show that a group $G$ is cyclic if and only if there exists a surjective group homomorphism from the additive group $\Z$ of integers to the group $G$.

Add to solve later 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.

Consider the ring

\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\]
and its ideal

\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\]
Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.

Suppose that $n\times n$ matrices $A$ and $B$ are similar.

Then show that the nullity of $A$ is equal to the nullity of $B$.

In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of $B$.

Let $p$ be a prime number. Let

\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.

Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.

Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ itself.

Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.

Add to solve later