Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Galous group, modules

  1. May 15, 2009 #1
    I was doing some self study and have questions:
    1. p(x)=x^{7}+11 over Q(a), R.
    where a is 7-th root of unity. What are Galouis groups?
    For the 1st case I got Z_{7}, second not sure. need hint for that
    2. need hint. I know it is easy: M is an R-module. Show that Hom_{R}(R,M)[tex]\cong[/tex]M.
    3. Spse that I is an ideal of R such that I^{k}=0 for some k>0 integer. Let M, N be R-modules and let [tex]\phi[/tex]:M->N be an R-module hom. Prove that if induced map [tex]\bar{\phi}[/tex]:M/IM->N/IN is surjective, then [tex]\phi[/tex] is surjective.
    4. show that 2[tex]\otimes[/tex]1 [tex]\neq[/tex]0 in 2Z[tex]\otimes[/tex]Z/2Z.
     
  2. jcsd
  3. May 16, 2009 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    2. You need a map from

    {f : R-->M } to M

    (or vice versa).


    Assuming that R is unital, then there is only one possible map you can write down:

    f ---> f(1)

    You have to try to show that is an isomorphism. You also may want to think about the other direction;

    M --> {f: R --> M}

    again, there is only one possible map you can write down - given m in M, then the only candidate in Hom_R(R,M) is translation by m:

    f_m(r)= r.m

    so you have to show that the map m--->f_m is an isomorphism (note we've dropped the explicit use use R being unital).
     
  4. May 16, 2009 #3
    This is what I tried. g(f)=f_{1}. But I had hard time showing that it is surjective
     
  5. May 16, 2009 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    Have you tried working to find an inverse? The second part of my hint wasn't just decoration.
     
  6. May 16, 2009 #5
    yeah, i got it. thanks
     
  7. May 17, 2009 #6
    #3 is somewhat tricky. The key is to prove that [tex] \varphi(IM) = IN [/tex]. This can be done by induction on [tex] k [/tex], but first you'll need a lemma to the effect that if [tex] \bar{\varphi} [/tex] is surjective, then the induced map [tex] \psi_r : I^r M/I^{r+1} \to I^r N/I^{r+1} [/tex] is surjective for all [tex] r [/tex].

    Let me think about the others some more...I'll get back to you in a bit. (Although #4 looks pretty trivial.)
     
    Last edited: May 18, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook