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 $F$.

Then if $a \in I$ is non-zero, we have that $1 = a^{-1} \cdot a \in I$, where $a^{-1}$ exists since $F$ is a field and $a \ne 0$.

Since $1 \in I$, for every element $b \in F$, $b = b \cdot 1 \in I$, so we have that $I = F = \ideal 1$ if $I \ne \set 0$.

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