Math 361, Spring 2019, Assignment 5

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

1. Canonical injection (of $R$ into $R[x]$).
2. Constant polynomial.
3. $D(x)$ (the field of rational expressions with coefficients in $D$).
4. Evaluation homomorphism (from $R[x]$ to $R$, sending $x$ to $a\in R$.)
5. Degree (of a polynomial).

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

1. Universal mapping property of $R[x]$ (for $R$ a commutative ring).
2. Bounds for the degrees of sum and product.
3. Theorem concerning the degree of a product, when the coefficient ring is an integral domain.
4. Theorem concerning the units of $D[x]$, when $D$ is an integral domain.
5. Theorem concerning zero-divisors in $D[x]$, when $D$ is an integral domain.

Solve the following problems:

1. Give an example of an infinite field of characteristic $p$.
2. Let $F$ be any field of characteristic $p$. Show that $F$ contains a subfield isomorphic to $\mathbb{Z}_p$. (Hint: consider the image of the initial morphism $\iota:\mathbb{Z}\rightarrow F$.)
3. The subfield you found above is called the prime subfield of $F$. Describe the prime subfield of the field you found in problem 1.
4. Now let $F$ be any field of characteristic zero. Show that $F$ contains a subfield isomorphic to $\mathbb{Q}$. (Hint: in this case the initial morphism $\iota:\mathbb{Z}\rightarrow F$ is injective. Now invoke the universal mapping property of $\mathrm{Frac}(\mathbb{Z})$.)
5. The subfield you found above is again called the prime subfield of $F$. Describe the prime subfield of $\mathbb{R}$.
6. Describe the prime subfield of $\mathbb{R}(x)$.
7. Section 22, problems 7, 9, 11, 13, and 25.

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