Math 361, Spring 2016, Assignment 1

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

1. Unit (of a unital ring).
2. Zero-divisor.
3. Field.
4. (Integral) domain.
5. Group of units (of a unital ring).
6. Euler totient function.

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

1. Theorem relating fields to integral domains.
2. Theorem concerning the group of units of a product ring.
3. Chinese Remainder Theorem (ring version).
4. Formula for $\phi(ab)$ when $a$ and $b$ are relatively prime.
5. Formula for $\phi(p^n)$ when $p$ is prime.
6. Euler's Theorem.
7. Fermat's Little Theorem.

Solve the following problems:

1. Section 20, problems 1, 5, 7, 9, and 10.

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

Solutions:

1. A ring is unital provided it has an identity. (Fun fact, the identity is called the unity, not unit) If an element of such a ring has a multiplicative inverse, that element is a unit.