Math 361, Spring 2020, Assignment 7

Read:

1. Section 22.

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

1. Polynomial expression (with coefficients in a commutative, unital ring $R$).
2. Sum (of two polynomial expressions).
3. Product (of two polynomial expressions).
4. $R[x]$.
5. Degree (of a polynomial expression; note our special convention regarding the degree of the zero polynomial).
6. Canonical injection (of $R$ into $R[x]$).
7. Constant polynomial.
8. Evaluation homomorphism (from $R[x]$ into $R$).

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

1. Inequality bounding the degree of a sum.
2. Conditions under which the above inequality is an exact equality.
3. Inequality bounding the degree of a product.
4. Conditions under which the above inequality is an exact equality.
5. Test for equality of two polynomial expressions.
6. Universal mapping property of $R[x]$.

Solve the following problems:

1. Section 22, problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 21, and 24.
2. Suppose $D$ is an integral domain, and $f$ is a unit in $D[x]$. Prove that $\mathrm{deg}(f)=0$. (Hint: use the formula for the degree of a product.)
3. Prove the Theorem characterizing the units of $D[x]$: the units of $D[x]$ are the units of $D$, regarded as constant polynomials.
4. Section 22, problem 25.
5. Suppose $R$ is a commutative, unital ring, and choose $a\in R$. Show that the evaluation homomorphism $\phi_a:R[x]\rightarrow R$ is never injective. (Hint: consider the polynomial $x-a$.)
6. Trying to generalize the previous problem to the more general evaluation homomorphisms described by the universal mapping property raises very subtle and interesting issues. Try to find specific rings $R,S$ and a homomorphism $\phi:R\rightarrow S$ and an element $a\in S$ for which $\phi_a$ is injective. (This is difficult, and the attempt will plunge you immediately into the icy waters of algebraic number theory. The classic example is $\phi:\mathbb{Q}\rightarrow\mathbb{R}$ with $a=\pi$ (the ratio of the circumference of a circle to its diameter). One says that $\pi$ is a transcendental number. We will have much more to say about this in coming lectures.)

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