Math 361, Spring 2019, Assignment 4

Read:

1. Section 21.
2. Section 22.

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

1. Fraction expression.
2. Equivalent (fraction expressions).
3. Fraction.
4. Sum (of two fractions).
5. Product (of two fractions).
6. $\mathrm{Frac}(D)$.
7. Canonical injection (of $D$ into $\mathrm{Frac}(D)$).
8. Polynomial function (on a ring $R$).
9. Polynomial (on a ring $R$).
10. Sum (of two polynomials).
11. Product (of two polynomials).
12. $R[x]$.

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

1. Statement relating $D$ to $\mathrm{Frac}(D)$ when $D$ is already a field.
2. Universal mapping property of $\mathrm{Frac}(D)$.

Solve the following problems:

1. Section 21, problems 1 and 2 (what is to be described in these problems is the image of the injection $\mu:\mathrm{Frac}(D)\rightarrow\mathbb{C}$ induced from the inclusion map $\iota:D\rightarrow\mathbb{C}$ via the universal mapping property; in other words, describe a "concrete model" of $\mathrm{Frac}(D)$ inside $\mathbb{C}$).
2. Section 22, problems 1, 3, 5, and 7.
3. Let $F$ be any field. We will show next week that the ring of polynomials $F[x]$ is an integral domain. Try, for various fields $F$, to understand what the field of fractions $\mathrm{Frac}(F[x])$ is like. That is, for each of several choices of $F$, write down several typical elements of $\mathrm{Frac}(F[x])$, then add and multiply them.

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