Math 361, Spring 2018, Assignment 2

Read:

1. Section 20.
2. Section 21.

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

1. Fraction (defined by two elements of an integral domain).
2. $\mathrm{Frac}(D)$, the field of fractions of the integral domain $D$.
3. Canonical injection (of $D$ into $\mathrm{Frac}(D)$).
4. Euler totient function.

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

1. Criterion for equality of fractions.
2. Cancellation law (for fractions).
3. Universal mapping property of $\mathrm{Frac}(D)$.
4. Chinese remainder theorem.
5. Theorem identifying the units of $\mathbb{Z}_n$.
6. Chinese remainder theorem.
7. Formula for $\phi(p^n)$.
8. Formula for $\phi(ab)$ when $\mathrm{gcd}(a,b)=1$.

Solve the following problems:

1. Section 20, problems 1, 3, 7, and 27.
2. Section 21, problem 1 (this problem asks you to find a "concrete model" for $\mathrm{Frac}(D)$, i.e. a subfield of $\mathbb{C}$ isomorphic to the field of fractions).
3. Section 11, problems 16, 18, and 20.
4. Prove Euler's theorem: if $a$ and $n$ are positive integers with $\mathrm{gcd}(a,n)=1$, then $a^{\phi(n)}\equiv1\ (\mathrm{mod}\ n)$. (Hint: the congruence class $[a]$ belongs to the group of units of $\mathbb{Z}_n$. Now use Theorem 10.12 on page 101.)

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