Math 361, Spring 2020, Assignment 6

Read:

1. Section 21.

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

1. Formal fraction (with numerator and denominator from an integral domain $D$).
2. Equivalence (of formal fractions).
3. Fraction (with numerator and denominator from $D$).
4. Sum (of two fractions).
5. Product (of two fractions).
6. Inverse (of a non-zero fraction).
7. $\mathrm{Frac}(D)$.
8. Canonical injection (of $D$ into $\mathrm{Frac(D)}$).
9. Concrete model (of $\mathrm{Frac}(D)$, arising from a particular injection

of $D$ into a particular field $F$).

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

1. Theorem concerning whether equivalence-of-formal-fractions is an equivalence relation.
2. Test for equality of fractions.
3. Theorem concerning whether addition and multiplication of fractions are well-defined operations.
4. Theorem characterizing which type of ring $\mathrm{Frac}(D)$ is.
5. Theorem concerning basic properties of fractions (please state at least a cancellation law, a description of the zero fraction, and two statements showing how a unit of $D$ can be "moved up and down" within a fraction).
6. Universal mapping property of the canonical injection $\iota:D\rightarrow\mathrm{Frac}(D)$.
7. Theorem describing $\mathrm{Frac}(F)$ when $F$ is already a field.

Solve the following problems:

1. Section 21, problems 1 and 2 (here "field of quotients" means "field of fractions," and you are being asked to describe the concrete model of $\mathrm{Frac}(D)$ arising from the inclusion $D\rightarrow\mathbb{C}$ (in problem 1) and the inclusion $D\rightarrow\mathbb{R}$ (in problem 2)).
2. Working in the field $\mathrm{Frac}(\mathbb{Z}_5)$, show that $\frac{2}{3}=\frac{4}{1}=\iota(4)$.
3. Working in $\mathrm{Frac}(D)$ for arbitrary $D$, show that $$\frac{a}{-b}=\frac{-a}{b}=-\frac{a}{b}.$$ (Hint: the second equality is proved by a different method than the first. To prove that two things are opposites, add them together and see what you get.)
4. Prove that every field of characteristic $p$ contains a copy of $\mathbb{Z}_p$, and that every field of characteristic zero contains a copy of $\mathbb{Q}$. (The copy of $\mathbb{Z}_p$ or $\mathbb{Q}$ that you find is called the prime subfield of $F$, and $\mathbb{Z}_p$ and $\mathbb{Q}$ are called the prime fields.) (Hint: if $\mathrm{char}(F)=p$ then the prime subring of $F$ is already a copy of $\mathbb{Z}_p$ and there is nothing to prove. If $\mathrm{char}(F)=0$, then apply the universal mapping property of $\mathrm{Frac}(\mathbb{Z})$ to the initial morphism $\mathbb{Z}\rightarrow F$.)
5. Is $\mathrm{Frac}(\mathrm{Frac}(D))$ an interesting object? Why or why not?

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