1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Characteristic of a field

  1. Apr 2, 2010 #1
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations



    3. 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.
     
  2. jcsd
  3. Apr 2, 2010 #2

    TMM

    User Avatar

    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?
     
  4. Apr 2, 2010 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If char(F)=m then doesn't that mean the field has an additive subgroup of order m?
     
  5. Apr 2, 2010 #4
    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)?
     
  6. Apr 2, 2010 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    A field is an additive group. The additive order of any element must divide the order of the field. Period.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Characteristic of a field
Loading...