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Showing a finite field has an extension of degree n

  1. Jul 16, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose [itex]F[/itex] is a finite field and [itex]n > 0[/itex].

    Show that [itex]F[/itex] has a field extension of degree [itex]n[/itex]


    2. Relevant equations

    Tower Law.


    3. The attempt at a solution

    Let [itex]p^m[/itex] be the size of [itex]F[/itex]

    It's trivial to note by characterization that there exists a finite field [itex]G'[/itex] of size [itex]p^{mn}[/itex] which has a subfield [itex]G[/itex] of size [itex]p^m[/itex].

    Then [itex]G \cong F[/itex]; by Tower Law, [itex][G' : G][/itex] is of degree n.

    I'm having trouble showing that there is an extension of [itex]F[/itex] that has degree n, however; I tried finding some isomorphism and using theorems about algebraic elements, but for some reason this stuff is not clicking in my head.

    I'm beginning to think the question only requires that I show that there is a field extension of an isomorphic copy, but that can't be right.

    Please give tiny hints only that can push me in the right direction or clarify some glaring error I've made.

    Thanks for your time.
     
  2. jcsd
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