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!

Homework Help: Finite field with prime numbers

  1. Nov 8, 2007 #1
    1. The problem statement, all variables and given/known data
    If F is a finite field show that there is a prime p s.t. (p times)a+a+...+a=0 for all a in the field

    2. Relevant equations

    3. The attempt at a solution
    Well I managed to prove that there must be an a in F s.t. (prime number, call p, times)a+a+...+a=0 but I can't seem to prove that for every a in F (p times)a+a+...+a=0 (This is the only approach I could think of).
  2. jcsd
  3. Nov 8, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    If you showed that pa=0 for some a, then pb = (pa)(a^-1 b) = 0 as well.

    Another approach goes as follows. Define a homomorphism f from Z (the ring of integers) into F by n -> n*1. Now Z/ker(f) is a subring of F, hence an integral domain, and consequently ker(f) is a prime ideal of Z. If F is finite, f cannot be an injection, so ker(f) isn't trivial, and is thus of the form pZ for some prime p. This means Z/pZ sits inside F, and in particular p=0 in F.
    Last edited: Nov 8, 2007
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook