Fields Revisited

A ring $(R, +, \cdot)$ is a field if $(R - \{0\}, \cdot)$ is an abelian group.

• this means there is a multiplicative identity $1 \neq 0$ and every nonzero element is a unit.

Alternatively, $(R, +, \cdot)$ is a field if

1. $(R, +)$ is an abelian gorup
2. $(R - \{0\}, \cdot)$ is an abelian group
3. The distributive law of $\cdot$ over $+$ holds.

Theorem If $(F, +, \cdot)$ is a field, then it has no proper divisors of zero.

• Proof

Ring properties of $\mathbb{Z}_n$

$(\mathbb{Z}_n, +, \cdot)$ is a ring, where both $+$ and $\cdot$ are modulo $n$.

It is in fact abelian under $\cdot$ as well as $+$.

Futhermore, it has a multiplicative identity, 1.

We know each $a \in \mathbb{Z}_n$ has a multiplicative inverse $a^{-1}$ iff $\gcd(a, n) = 1$

Hence in $\mathbb{Z}_n$, $[a]$ is a unit iff $\gcd(a, n) = 1$

Theorem $\mathbb{Z}_n$ is a field iff $n$ is a prime.

• Proof

Polynomials

Let $(R, +, \cdot)$ be a ring.

Let $x$ denote an indeterminate (a formal symbol that is not an element of $R$)