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 $(\mathbb{Q}, +, \cdot)$, the element $\frac{19}{1}$ is a unit, because it has inverse $\frac{1}{19}\in\mathbb{Q}$.

   $\neg$e.g. In $(\mathbb{Z}, +, \cdot)$, the element $19$ is not a unit, because nothing multiplied by $19$ can produce the identity.

2. 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 $0$)

   e.g. In $(\mathbb{Z}_4, +, \cdot)$, $2$ is a zero divisor, because $2\cdot2=4=0$.

   $\neg$e.g. In the same ring, $3$ is not a zero divisor: $3\cdot1\neq0$, $3\cdot2\neq0$, $3\cdot3\neq0$.

3. A field is a unital ring of which every nonzero element is a unit (note: this property alone defines a division ring), the second binary operation ("multiplication") of which is commutative.

   e.g. $(\mathbb{Q}, +, \cdot)$

   $\neg$e.g. $(\mathbb{Z}, +, \cdot)$

4. An integral domain (often referred to as domain) is a commutative, unital ring with no zero divisors.

   e.g.Any field is a domain.

   $\neg$e.g. From the top example in number 2, $(\mathbb{Z}_4)$ is almost a domain, if it weren't for that pesky $2$.
5. The units of a unital ring form a group under multiplication. This group is called the group of units.

   e.g. Back in $(\mathbb{Z}_4, +, \cdot)$, the units are $1$ ($1\cdot1=1$) and $3$ ($3\cdot3=1$), and so the group of units of $(\mathbb{Z}_4, +, \cdot)$ is the group $(\{1, 3\}, \cdot)$.

   $\neg$e.g. The group of units of $(\mathbb{Z}_4, +, \cdot)$ is not $(\{0, 1, 3)\}, \cdot)$, because $0$ is not a unit.

6. The Euler totient function is the function $\phi:\mathbb{Z}^+\to\mathbb{Z}^+$, whose output for $n\in\mathbb{Z}^+$ is the cardinality of the group of units of $(\mathbb{Z}_n, +, \cdot)$.

   e.g. $\phi(5)=4$, because the units of $(\mathbb{Z}_5, +, \cdot)$ are $1, 2, 3$ and $4$. Those are four numbers $\Box$

   $\neg$e.g. $\phi(4)=3$ is not a true statement. As discussed earlier, the units of $(\mathbb{Z}_4, +, \cdot)$ include $1$ and $3$. Those are two numbers, not enough to be three numbers.