Is there any relationship between Simple Group and Field?

Question

I have studied definition of simple group. I know the fact that concept of normal subgroup is analogous to ideal in ring theory. A field is a ring which do not have any nontrivial ideal. A simple group do not have any nontrivial normal subgroup. So, I have 2 questions 1) Is field and simple group are 2 equivalent concepts? 2) Does every field is a simple group under addition?

Answer 1

They are very different concepts. They are similar in that they are both objects with no non-trivial quotients, but they are not the same thing at all.

A field is only a simple additive group if it is exactly $\mathbb{Z}/p\mathbb{Z}$, as those are the only abelian simple groups.