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Math Help - subfields of a field

  1. #1
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    subfields of a field

    Identify all the subfields of a field F=F_{p^{18}}, with p^{18} elements where p 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.
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  2. #2
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    Quote Originally Posted by numbertheory12 View Post
    Identify all the subfields of a field F=F_{p^{18}}, with p^{18} elements where p 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.
    The subgroups of \mathbb{F}_{p^{18}} are \mathbb{F}_{p^{k}} where k|18. Furthermore, \mathbb{F}_{p^{k}}\subseteq \mathbb{F}_{p^{j}} if and only if k|j.
    Now you can draw the diagram of all subfields.
    Hint: The subfield diagram should look like an inverted \mathbb{Z}_{18} diagram*.


    (*This is not an accident! It follows from fundamental theorem of Galois theory.)
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  3. #3
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    I got this (I don't know how draw lattice diagrams in latex): Is that correct?
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  4. #4
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    Quote Originally Posted by numbertheory12 View Post
    I got this (I don't know how draw lattice diagrams in latex): Is that correct?
    I do not see any picture.
    You can put it into Paint and then upload it to MHF.
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  5. #5
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    http://lh5.ggpht.com/_miqRA1rlzTU/ST...dpZsJ6Fs/1.jpg

    Here it is. I am pretty sure it is correct though. Thanks for the help.
    Attached Thumbnails Attached Thumbnails subfields of a field-1.bmp  
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  6. #6
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    Good job. You just missed one more field, \mathbb{F}_p, the base field.
    It goes below \mathbb{F}_{p^3} and \mathbb{F}_{p^2}.
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