1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook