Field is Principal Ideal Domain

Theorem

Let $F$ be a field.

Then $F$ is a principal ideal domain.

Proof

Let $F$ be a field.

Let $I \subset F$ be a non-null ideal of $F$.

Let $a \in I$ be non-zero.

Since $F$ is a field, $a^{-1}$ exists.

We have that $1 = a^{-1} \cdot a \in I$.

Since $1 \in I$, for every element $b \in F$:

$b = b \cdot 1 \in I$

we have that $I = F = \ideal 1$ if $I \ne \set 0$.

Thus the only ideals of $F$ are $\ideal 0 = \set 0$ and $\ideal 1 = F$, which are both principal ideals.

Hence $F$ is a principal ideal domain.

$\blacksquare$