Ideals of Field

Theorem
Let $$\left({F, +, \circ}\right)$$ be a field whose zero is $$0_F$$.

The only ideals of $$\left({F, +, \circ}\right)$$ are $$\left\{{0_F}\right\}$$ and $$F$$ itself.

Proof
By definition, a field is a division ring.

The result follows directly from Ideals of a Division Ring.