Math 361, Spring 2022, Assignment 8

Read:

1. Section 21.
2. Section 22.

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

1. Formal fraction (from an integral domain $D$).
2. Equivalence (of formal fractions).
3. Fraction (from an integral domain $D$).
4. Addition (of fractions).
5. Multiplication (of fractions).
6. $\mathrm{Frac}(D)$ (the field of fractions of the integral domain $D$).
7. Canonical injection (of an integral domain $D$ into its field of fractions).
8. Polynomial function (from a ring $R$ into itself).
9. Polynomial expression (with coefficients in a ring $R$).
10. Addition (of polynomial expressions).
11. Multiplication (of polynomial expressions).
12. $R[x]$ (the ring of polynomial expressions, with coefficients in $R$, in the indeterminate $x$, or "$R$ adjoin $x$" for short).

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

1. Equality test for fractions.
2. Universal mapping property of $\mathrm{Frac}(D)$.
3. Example of two distinct polynomial expressions that give rise to the same polynomial function.

Solve the following problems:

1. Section 21, problems 1 and 2 (translation of problems: you are being asked to describe the concrete model of the field of fractions of the given integral domain $D$ arising from the inclusion of $D$ in the given field $F$).
2. Section 22, problems 1, 2, 3, 4, 5, and 6.
3. (Rational expressions). Next week we shall prove that whenever $D$ is an integral domain, so is $D[x]$. For purposes of this exercise, you may take this fact for granted. Thus, the field of fractions of $D[x]$ is a well-defined object, which is usually denoted $D(x)$. Write down two "random" elements of the field $\mathbb{R}(x)$, and show how to add them, and also how to multiply them.
4. (An infinite ring with positive characteristic). Let $R=\mathbb{Z}_3[x]$ denote the ring of polynomial expressions with coefficients in $\mathbb{Z}_3$. Write the table of values of the initial morphism $\iota:\mathbb{Z}\rightarrow R$, and show that $\mathrm{char}(R)=3$.
5. Let $R$ be as in the previous exercise. Show that $R$ is an infinite ring, even though it has characteristic three and its prime subring is thus a copy of $\mathbb{Z}_3$.
6. Let $R$ be as in the previous exercise and put $F=\mathrm{Frac}(R)$. (We will show next week that $R$ is an integral domain; for purposes of this problem you may take this for granted.) Show that $F$ is an infinite field of positive characteristic.

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