For each part, compute the sum and product of the given polynomials in the given polynomial ring.
$f(x) = 1 + 2x$ and $g(x) = 1 + 3x$ in $\mathbb{Z}_6[x]$.
$f(x) = 1 + 2x + 3x^2$ and $g(x) = 4x + x^3$ in $\mathbb{Z}_7[x]$.
Let $R$ be a ring and let $f(x),g(x) \in R[x]$.
Assuming that $R$ is an integral domain, prove that $\deg f(x)g(x) = \deg f(x) + \deg g(x)$.
Hint: split into cases depending on whether one or both is the zero polynomial. Recall that the degree of the zero polynomial is $-\infty$ and we define $(-\infty) + n = n + (-\infty) = -\infty$ for all $n$.
Explain why one of your answers to problem 1 shows that the hypothesis that $R$ is an integral domain is necessary in part a.
Let $R$ be a ring. Prove that if $R$ does not have unity, then $R[x]$ does not have unity.
Hint: prove the contrapositive. Assume that $R[x]$ has a unity $f(x)$ and consider the coefficient of $x^0$ in $f(x)$.
Let $p$ be a prime. Prove that there is an infinite field of characteristic $p$.
Hint: consider a field of quotients of some ring.
Let $F'$ be a field and let $F$ be a subfield of $F'$. Let $\alpha \in F'$ and consider the evaluation homomorphism $\phi_\alpha : F[x] \to F'$. Prove that the following two statements are equivalent:
The kernel of $\phi_\alpha$ contains a polynomial of degree $1$.
$\alpha \in F$.
An automorphism of a ring is an isomorphism from that ring to itself. An automorphism is nontrivial if it is not the identity function. Prove that the polynomial ring $\mathbb{Z}[x]$ has a nontrivial automorphism.
Hint: consider an evaluation homomorphism $\phi_\alpha$ where $\alpha \in \mathbb{Z}[x]$. (Yes, you can "evaluate" a polynomial at another polynomial!) Which $\alpha$ will work?