Field is Principal Ideal Domain
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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.
$\blacksquare$