Ideals of Field

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

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

Proof

 * From Null Ideal, $$\left\{{0_R}\right\}$$ is an ideal of $$\left({F, +, \circ}\right)$$, as $$\left({F, +, \circ}\right)$$, being a field, is also a ring.


 * By definition, a field is a division ring.

By definition, every non-zero element of a division ring is a unit.

So $$S \ne \left\{{0_F}\right\} \Longrightarrow \exists f \in S: f \ne 0_F$$ such that $$f$$ is a unit of $$F$$.

The result follows from Ideal of Unit is Whole Ring.