Math 361, Spring 2016, Assignment 3

Carefully define the following terms, then give one example and one non-example of each:

1. Polynomial ring (i.e. $R[x]$ where $R$ is any commutative unital ring).
2. Indeterminate (i.e. the $x$ in $R[x]$).
3. Multivariate polynomial ring (i.e. $R[x_1,\dots,x_n]$).
4. Field of rational expressions (i.e. $D(x_1,\dots,x_n)$). (Why not $R(x_1,\dots,x_n)$?)
5. Constant polynomial.
6. Degree (of a univariate polynomial).
7. Evaluation homomorphism.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning degrees of sums and products.
2. Theorem concerning the units of $D[x]$ (where $D$ is a domain).
3. Theorem concerning zero-divisors in $D[x]$.
4. Theorem concerning extension of homomorphisms to polynomial rings (i.e. the theorem that starts "Given any homomorphism $\phi:R\rightarrow S$ and any fixed element $c\in S$, there exists one and only one...").
5. Theorem concerning polynomial long division.

Solve the following problems:

1. Section 22, problems 7, 13, and 25(b-c).
2. Section 23, problems 1 and 3.
3. Show that $1+2x$ is a unit in $\mathbb{Z}_4[x]$. Does this contradict the theorem concerning units of $D[x]$? Does it contradict the theorem concerning the degrees of sums and products?

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