• Support PF! Buy your school textbooks, materials and every day products Here!

Characteristic of a field

  • Thread starter dmatador
  • Start date
  • #1
120
1

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.
 

Answers and Replies

  • #2
TMM
92
0
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?
 
  • #3
Dick
Science Advisor
Homework Helper
26,258
618
If char(F)=m then doesn't that mean the field has an additive subgroup of order m?
 
  • #4
120
1
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)?
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
618
A field is an additive group. The additive order of any element must divide the order of the field. Period.
 

Related Threads for: Characteristic of a field

  • Last Post
Replies
1
Views
2K
Replies
4
Views
3K
  • Last Post
Replies
0
Views
642
  • Last Post
Replies
3
Views
522
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
1K
Top