Characteristic of a field

In summary, the conversation discusses the proof that for a field with order 2^n, the characteristic must be 2. The reasoning involves using Lagrange's theorem and considering the subfield generated by 1. It is concluded that the order of any element in the field must divide 2^n, resulting in a characteristic of 2.

Homework Statement

Let F be a field with order 2^n. Prove that char (F) = 2.

The Attempt at a Solution

My reasoning is that since a field is an integral domain, its characteristic must be either 0 or prime. After that I get confused, because would the char (F) need to somehow be related to the order of the field? Is there some reasoning that since it must divide the order of the field (just spit balling) and it must be prime, that it could just be 2? I know this is by no means a proof, but I am having difficulty finding some strong ideas to finish this.

Consider the subfield generated by 1. Its order must divide the order of the field, by Lagrange's theorem, since a field is an additive group. What does this tell you?

If char(F)=m then doesn't that mean the field has an additive subgroup of order m?

TMM said:
Consider the subfield generated by 1. Its order must divide the order of the field, by Lagrange's theorem, since a field is an additive group. What does this tell you?

so the order of any element of the field must divide 2^n... so it should be a number of the form 2m (m being an integer)?

A field is an additive group. The additive order of any element must divide the order of the field. Period.