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!

Product Field

  1. Aug 15, 2010 #1
    "Product Field"

    1. The problem statement, all variables and given/known data
    Let K and L be subfields of a field M such that M/K (the field extension M of K) is finite. Denote by KL the set of all finite sums ∑xiyi with xi ∈ K and yi ∈ L. Show that KL is a subfield of M, and that

    [KL : K] ≤ [L : K ∩ L].


    2. Relevant equations
    [KL : K] etc. is the standard notation for the dimension of KL as a vector space over K (includes infinity).


    3. The attempt at a solution

    I've done the first part about the subfield fine, it's the inequality I'm struggling with. I've tried using the tower law ([K:M]=[K:L][L:M] where M is a subfield of L is a subfield of K. However, I can't seem to appropriately 'link up' the right subfields to relate the left-hand and right-hand side of the inequality; despite this, since it's an inequality I'm not even convinced the tower law is the right way to go: perhaps there's a nicer way to solve the problem, like finding an isomorphism between KL/K and a subfield of L/(K ∩ L) or something similar - but I have a habit of overthinking things, so maybe I'm overdoing it!

    Any suggestions would be muchly appreciated - thanks in advance :)
     
  2. jcsd
  3. Aug 15, 2010 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: "Product Field"

    The tower law probably isn't the way to go.

    Looking at injections is a good idea. It might help to start with a small step: Can you see why L is a spanning set of vectors for KL over the field K? From there, think about how you could pare that down to a basis, and in particular why the elements of L∩K for the most part will not be included
     
  4. Aug 15, 2010 #3
    Re: "Product Field"

    I've just noticed also this looks a lot like the second isomorphism theorem - perhaps another way to go about the problem?

    Anyway, i can see why L is a spanning set of vectors for KL over K, and I guess you could use AoC to pick a basis by selecting an element of L at random, then another not in its spanning set over K, then another not in the spanning set of the first 2 elements picked over K, and so on. I'm not completely sure why the intersection won't be included - I can see vaguely that anything in the intersection will have an inverse in K, so can be 'included' in the K-term (i.e. k*1, for some k in K) in any basis expansion, but I don't think I really understand properly. Could you elaborate please? Thanks very much.
     
  5. Aug 15, 2010 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: "Product Field"

    You can find a basis of KL over the field K containing only elements in L.

    What do those elements do when you just look at them as elements of L over the field L∩K?
     
  6. Aug 16, 2010 #5
    Re: "Product Field"

    I'm still not completely sure sorry, I'm obviously not getting this :( When you're looking at L over L∩K, the span of your elements will have to be smaller than or equal to the span of those elements over K, but I can't really see how to formulate this idea properly - sorry to keep asking! I'm very new to Galois theory and all I know about it is currently self taught, so unfortunately it's taking me a while from time to time to get my head around things - the help is greatly appreciated.
     
  7. Aug 16, 2010 #6

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Re: "Product Field"

    We're not really interested in the span. We have a basis for KL in terms of elements of only L. I claim those elements are linearly independent vectors in L as a vector space over the subfield L∩K.

    Do you see why that would wrap up the proof?
     
  8. Aug 16, 2010 #7
    Re: "Product Field"

    Ah of course, it makes perfect sense when you put it like that :) The argument is fairly simple once you spot it, I was definitely overcomplicating things - thankyou for being so patient!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook