1. Not finding help here? Sign up for a free 30min 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!

Question on normal field ext.

  1. Dec 12, 2007 #1
    If g(x)[tex]\in[/tex] K[x] and 1< deg(g)=n.
    Given that G/K is a normal field ext., if g(x)=g1(x)*....*gk(x)[tex]\in[/tex] G[x],
    then deg(g1)=....=deg(gk)

    My attempt:
    I let G = K adjoins the coefficients of gi's.
    Let [tex]\alpha[/tex] be a root of g.
    Notice that K [tex]\subseteq[/tex] G [tex]\subseteq[/tex] K( [tex]\alpha[/tex]) = G( [tex]\alpha[/tex])

    We can express g(x) = irr( [tex]\alpha[/tex], G) *p(x) for some p(x) [tex]\in[/tex] K[x].
    if i let g1 be irr( [tex]\alpha[/tex], G),
    then deg(g1) = deg( irr( [tex]\alpha[/tex], G)) = [G([tex]\alpha[/tex]):G]=[K([tex]\alpha[/tex]):G]

    Next, we do the same thing again with another root, say [tex]\beta[/tex].
    g(x) = irr([tex]\beta[/tex], G) *q(x) for some q(x) [tex]\in[/tex] K[x]
    if i let g2 be irr( [tex]\beta[/tex], G),
    then deg(g2) = deg( irr( [tex]\beta[/tex], G)) = [G([tex]\beta[/tex]):G]=[K([tex]\beta[/tex]):G]

    if we proceed in the same way, deg(gi) can be found to be equal to [K([tex]\theta[/tex]):G] for some root [tex]\theta[/tex] of g.

    Next, we observe that since K([tex]\alpha[/tex]) and K([tex]\beta[/tex]) are normal extnesion of K, they split into linear factors for g(x). so K([tex]\alpha[/tex]) = K([tex]\beta[/tex]) (and this in fact implies to K adjoining other roots of g)

    so deg(g1)=......=deg(gk)

    please tell me if there's anything wrong with it.. this question is a bit too advanced for me :confused:
  2. jcsd
  3. Dec 12, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    I suppose we're assuming that g(x) is irreducible over K. I skimmed through your solution, and I think you have the right ideas, but your write-up is very hard to read - and some things don't make sense, e.g. what do you mean when you say "K(a) splits into linear factors for g(x)"? If I'm not mistaken, this problem is from Chapter V of Hungerford's Algebra, correct? It was one of my favorite problems!

    Here are some tips that will help make your post more readable: When you want to use inline TeX (i.e. during sentences), use [itex] instead [tex]. This will make everything align nicely. Also, try to state clearly what you're trying to show at each step.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Question on normal field ext.