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Subfields of a field

  1. Dec 9, 2008 #1
    Identify all the subfields of a field [tex]F=F_{p^{18}}[/tex], with [tex]p^{18}[/tex] elements where [tex]p[/tex] is a prime. Draw the lattice of all subfields.

    I am allowed to just use a theorem, but I don't know one to use.
  2. jcsd
  3. Dec 9, 2008 #2


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    Start with Lagrange's theorem (which is for groups, but all fields are additive groups) to get that the order of any subgroup (and hence the order of any subfield) divides p18. This narrows down the possible orders of the fields you're looking for
  4. Dec 9, 2008 #3
    Also, if you have a subfield K, its group of units K* is a subgroup of F* as well.

    Don't you have a theorem that says the subfields of Fpn are exactly Fpm where m divides n?

    This stuff belongs in the Abstract Algebra forum.
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