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:

Vocab

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.

   e.g. In $\mathbb{Q}$, the element $\frac{19}{1}$ is a unit, because it has inverse $\frac{1}{19}\in\mathbb{Q}$.
   $\neg$e.g. In $\mathbb{Z}$, the element $19$ is not a unit, because nothing multiplied by $19$ can produce the identity.

1. A zero divisor is a nonzero element $a$ such that $\exists b \neq 0$ with $ab = 0$ or $ba = 0$. (The kicker being you can get the zero element without actually multiplying by $0$)

1. A division ring is a unital ring of which every nonzero element is a unit.