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: Extension field

  1. Feb 13, 2008 #1
    [SOLVED] extension field

    1. The problem statement, all variables and given/known data
    Let E be an extension field of Z_2 and [itex]\alpha[/itex] in E be algebraic of degree 3 over Z_2. Classify the groups [itex]<Z_2(\alpha),+>[/itex] and [itex]<Z_2(\alpha)^*,\cdot>[/itex] according to the fundamental theorem of finitely generated abelian groups.
    [itex]Z_2(\alpha)^*[/itex] denotes the nonzero elements of Z_2(\alpha).

    2. Relevant equations

    3. The attempt at a solution
    The first group is obviously Z_2 cross Z_2 cross Z_2, right? I am using that theorem that says that every element of F(\alpha) can be uniquely expressed as a polynomial in F[\alpha] with degree less than 3. I am so confused about how to find the second group since they didn't give me explicitly the irreducible polynomial for [itex]\alpha[/itex] over F? Is the problem impossible?
    Last edited: Feb 13, 2008
  2. jcsd
  3. Feb 13, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    The first group is indeed (Z_2)^3.

    As for the second one: How many elements are in (Z_2(alpha))*?
  4. Feb 13, 2008 #3
    8-1=7, so it has to be Z_7!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook