Polynomial Rings

Let $R$ be a commutative ring containing an element $a$. Let $p(x)\in R[x]$. The evaluation homomorphism is the map $\phi_{a}:R[x]\to R$ where $\phi_{a}(p(x))=p(a)$. Prove that $\phi_{a}$ is indeed a ring homomorphism.

Prove that polynomial multiplication is associative.(Hint: $\displaystyle c_{k}=\sum_{i+j=k}a_{i}b_{j}$, which is the sum over all non-negative integers $i$ and $j$ with the property that $i+j=k$.)

Show that $\mathbb{Q}[\sqrt[3]{2}]=\left\{a+b\sqrt[3]{2}+c\sqrt[3]{4}\mid a,b,c\in\mathbb{Q}\right\}$.

Find a degree 5 polynomial in the ring $\mathbb{Z}_{2}[x]$.

Find the zeroes of the polynomial $x^{2}-1\in\mathbb{Z}_{4}[x]$.

Let $R$ be a commutative ring.

1. Show that any element in $R$ is a constant polynomial of $R[x]$.

2. If $f(x)$ is a constant polynomial in $R[x]$, show that $f(x)\in R$.

Let $R$ be a ring.

1. Prove that $R$ is a ring with identity if and only if $R[x]$ is a ring with identity.

2. Prove that $R$ is a commutative ring if and only if $R[x]$ is a commutative ring.

Show that the map $\phi$ in 1.33 is a ring homomorphism.

Divide $3x^{4}+3x^{3}+4x^{2}+3x+3$ by $x^{2}+2x+3$ in $\mathbb{Z}_{5}[x]$ and hence state two factors of $3x^{4}+3x^{3}+4x^{2}+3x+3$ in $\mathbb{Z}_{5}[x]$.

Find a degree 4 polynomial that is equal to the polynomial $1-2x+3x^{2}-4x^{3}+5x^{4}-6x^{5}$ in $\mathbb{Z}_{3}[x]$.

Let $I=\left\langle x^{2}+3x-1\right\rangle$. Simplify

in the quotient ring $\mathbb{Z}[x]/I$.

For the polynomials

1. $4x^{2}+2x+1\in\mathbb{Z}_{8}[x]$ and

2. $x^{2}+2x+1\in\mathbb{Z}_{7}[x]$,

find another polynomial $f(x)$ such that their product is 1 for all $x$ in the ground ring. If such an $f(x)$ does not exist, explain why.

Prove the remainder theorem (1.41).

Let $I=\{f(x)\in\mathbb{Z}[x]\mid f(-2)=0\}$ be a subset of $\mathbb{Z}[x]$. Consider the homomorphism $\phi:\mathbb{Z}[x]\to\mathbb{Z}$ where $\phi(f(x))=f(-2)$ for all $f(x)\in\mathbb{Z}[x]$.

1. Show that $I$ is an ideal of $\mathbb{Z}[x]$.

2. Hence determine if the ideal $I$ is prime, maximal, or both.

Show that $\left\langle x\right\rangle$ is a prime ideal in $\mathbb{Z}[x]$.

Prove that $\mathbb{Z}[x]/\left\langle x\right\rangle\cong\mathbb{Z}$.

Find an infinite set of polynomials $S\subseteq\mathbb{Z}_{5}[x]$ such that any two distinct polynomials in $S$ have different degrees and all of $\mathbb{Z}_{5}$ are zeroes of any polynomial in $S$.

Let $F$ be a field and let $I$ be a non-zero ideal in $F[x]$.

1. If $g(x)\in I$ is a non-zero polynomial of minimum degree, prove that $I=\left\langle g(x)\right\rangle$.

2. If $I=\left\langle g(x)\right\rangle$ for a $g(x)\in I$, prove that $g(x)$ is non-zero and of minimum degree.

Prove $R[x]\cong R[x^{k}]$ for any positive integer $k$ and commutative ring $R$.

Let $R$ and $S$ be commutative rings, and let $\phi:R\to S$ be a ring homomorphism. Define the map $\psi:R[x]\to S[x]$ where

1. Show that $\psi$ is a ring homomorphism.

2. If $\phi$ is an isomorphism, prove that $R[x]\cong S[x]$.

Prove that $\mathbb{Z}[x]$ is not a PID.(Hint: consider the set $\{2f(x)+xg(x)\mid f(x),g(x)\in\mathbb{Z}[x]\}$.)