Math 361, Spring 2017, Assignment 1

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

1. Group of units (of a unital ring).
2. Direct product (of two rings).
3. Euler totient function.

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

1. Theorem characterizing the units of $\mathbb{Z}_n$.
2. Formula for $\phi(p^n)$ when $p$ is prime.
3. Theorem relating $G(R\times S)$ to $G(R)\times G(S)$.
4. Formula for $\phi(ab)$ when $a$ and $b$ are relatively prime.
5. Euler's Theorem.
6. Fermat's Theorem (this is a special case of Euler's theorem; you will find it on page 184 of the text).

Solve the following problems:

1. Section 20, problems 1, 3, 5, 7, and 10.

--------------------End of assignment--------------------