Math 361, Spring 2017, Assignment 3

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

1. Polynomial function defined by a polynomial expression.
2. Zero (a.k.a. root) of a polynomial.
3. Degree (of a polynomial).
4. $R[x, y]$.
5. $R[x_1,\dots,x_n]$.

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

1. Bound on the degree of the sum of two polynomials.
2. Bound on the degree of the product of two polynomials.
3. Exact formula for the degree of the product of two polynomials, in case the base ring is an integral domain.

Solve the following problems:

1. Section 22, problems 5, 7, 9, 11, 13, and 15.
2. Prove that if $D$ is an integral domain, then so is D[x]. (Hint: the only hard part is showing that the product of two non-zero polynomials is non-zero. So suppose $f,g\neq0$. What can you say about $\mathrm{deg}(f)$, $\mathrm{deg}(g)$, and $\mathrm{deg}(fg)$? Alternatively, what can you say about the highest-degree term of $fg$?)
3. Let $D$ be any integral domain. The field of rational expressions in one indeterminate, with coefficients in $D$ is the field $D(x) = \mathrm{Frac}(D[x])$. Write down two or three sample elements of $\mathbb{Q}(x)$, then show how to add, multiply, and invert them.
4. More generally, the field of rational expressions in $n$ indeterminates is the field $D(x_1,\dots,x_n) = \mathrm{Frac(D[x_1,\dots,x_n])}$. Give sample elements and sample calculations, as above, for the field $\mathbb{Q}(x,y)$.

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