What is the Characteristic of a Field with Order 2^n?

[dmatador]

Homework Statement

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

Homework Equations

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.

 

[TMM]

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?

 

[Dick]

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

 

[dmatador]

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)?

 

[Dick]

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