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Math Help - Order of a finite field

  1. #1
    Junior Member
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    Order of a finite field

    Want to show a finite field has order p^n, where p is prime

    ATTEMPT:

    Let F be a field, then char(F) = p, a prime (Using some theorems in my algebra book). So p*1 = 1+1+...+1=0 so |1|=p. Then for any r in F p*r = r+r+...+r=r*(1+1+...+1)=0 implies |r| = p. so p divides |F|

    Now assume that q is a prime, q not equal to p and q divides |F|. By Cauchy's theorem for finite abelian groups, F then has an element of order q, which is a contradiction that all elements have order p.


    I do not know where I can go from there...Can I just say that |F| = p^n? I do not see that. THANKS!!!
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  2. #2
    Senior Member roninpro's Avatar
    Joined
    Nov 2009
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    I'll sketch an outline and let you fill in the details.

    1) Show that F has a subfield isomorphic to \mathbb{Z}/p\mathbb{Z}
    2) Claim that F is a field extension of \mathbb{Z}/p\mathbb{Z} of finite degree
    3) Take a basis for F over \mathbb{Z}/p\mathbb{Z} and count the possible number of linear combinations of the basis vectors and conclude
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