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Homework Help: Galois Theory - Algebraic extensions

  1. Apr 1, 2010 #1
    1. The problem statement, all variables and given/known data

    Let M/L and L/K be algebraic field extensions. Is M/K necessarily algebraic?

    2. Relevant equations

    Tower law: [M:K]=[M:L][L:K]

    3. The attempt at a solution

    If both M/L and M/K are finite extensions then by the tower law M/K is also a finite extension, hence is algebraic. So one or both of them must be infinite. The only infinite algebraic extensions I can think of are similar in construction to the algebraic closure of the rationals.
    An element [tex]m \in M[/tex] is algebraic over L so we can write [tex]\sum a_{i}m^i=0[/tex] for some [tex]a_i \in L[/tex], where i runs from 0 to some n. Might be able to use the fact that L/K is algebraic now?

    Thanks for any help!
  2. jcsd
  3. Apr 1, 2010 #2
    You have now singled out an element m and want to show that it's algebraic over K. Consider the tower:
    [tex]K \subseteq K(a_1,\ldots,a_n) \subseteq K(a_1,\ldots,a_n,m)[/tex]
    You can show that m is algebraic over [itex]K(a_1,\ldots,a_n)[/itex] and you should therefore be able to get back to your finite case.
  4. Apr 1, 2010 #3
    Funnily enough I was just working via this argument on paper to see whether I could delete the thread. At least now I know I was on the right lines, thanks very much!
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