Ideals in a real/complex number field? Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$.
Quick web search gave no satisfactory results. Yet I believe this couldn't be considered trivial, could it?
I am contemplating quite a few ideas for example looking for possible kernels of homomorphisms. Maybe you could give some references? Ideally I'd love to figure out even multiple different proofs for this.
 A: Fields only have the two obvious ideals. That is because in a field, every non-zero element is invertible, and invertible elements generate the entire ring as ideals.
A: I think you are getting confused by the terminolgy. Neither ${\mathbb R}$ nor ${\mathbb C}$ is a number field. A number field is a subfield of ${\mathbb C}$ which is a finite extension of ${\mathbb Q}$, such as ${\mathbb Q}(i)$ or ${\mathbb Q}(\sqrt[3]{2})$.
As others have pointed out, fields only have two ideals. But every number field $K$ has a ring of integers $D$ (which is the subring of the algebraic integers in the field), and it is customary within the algebraic number theory community to refer to ideals of $D$ as ideals of the number field. To further complicate matters, there are fractional ideals, which are finitely generated $D$-submodules $M$ of $K$ with the property that $cM \le K$ for some $c \in D$, and these are sometimes referred to simply as ideals.
A: It is known that a ring $R$ is a field iff it contains only trivial ideals, i.e., the only ideals in $R$ are $\{0\}$ and $R$ itself. See, e.g., this post.
