Why any field is a principal ideal domain?
According to the definition of P.I.D, first, a ring's ideal can be generated from a single element; second, this ring has no zero-divisor. This two conditions make a ring P.I.D.
But how to prove any field is P.I.D?
Best Answer
Let $F$ be a field and $I \subset F$ be a nontrivial ideal. Then if $a \in I$ is nonzero, we have that $1 = a^{-1} \cdot a \in I$, where $a^{-1}$ exists since $F$ is a field and $a \neq 0$. Since $1 \in I$, for every element $b \in F$, $b = b \cdot 1 \in I$, so we have that $I = F = \langle 1 \rangle$ if $I \neq \{0\}$.
In conclusion, the only ideals of a field $F$ are $\langle 0 \rangle = \{0\}$ and $\langle 1 \rangle = F$, which are both principal ideals.