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

Struggling to understand a field theorm's corollary

  1. Feb 27, 2015 #1
    Theorem: Let F be any field. If G is a finite subgroup of the multiplicative group F* of F, then G is cyclic. In particular, is F is finite, then F* is cyclic.

    Corolarry 1: GF(p^n) = Z_p(u), where u is any primitive element for GF(p^n).

    So <u> = GF(p^n)*, so |u| = GF(p^n) - 1.

    I'm now trying to imagine what Z_p(u) would look like, maybe:

    {a_0 + a_1*u + a_2*u^2 + ...... a_(n-1)*u^n-1 | a_i are elements of Z_p, u^n = (0?)}

    This would make sense because this field would have order n... but it would also mean that u^n = (0?) or something, when u^n should just equal u^n because |u| = GF(p^n) - 1 > n.

    Anyone understand my dilemma? If anyone could drop some knowledge on this topic in general it'd be appreciated.
     
  2. jcsd
  3. Feb 27, 2015 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Well, let's look at an example. Take ##\mathbb{F}_4 = \{0,1,a,a+1\}##, where ##a^2 = a+1##. The primitive element is clearly ##a## and this has order ##3## because
    [tex]a^2 = a+1~\text{and}~a^3 = a(a+1) = a^2 + a = a+1+a =1[/tex]
    We indeed have ##\mathbb{F}_4 = \{\alpha + \beta a~\vert~\alpha,\beta\in \mathbb{Z}_p\}##. But you see that this does not imply that ##a^2 = 1##. The issue is that ##\alpha + \beta a## and ##a^n## are very different notations which might coincide. In this situation, we have ##a^2 = a+1##. So if you look at the cyclic element ##a##, then we have a representation ##\{0,a,a^2,a^3\}## and when you look at it your way then we have ##\{0,1,a,1+a\}##. These are two very different notations.

    I encourage you to try other examples to see this more clearly.
     
  4. Feb 27, 2015 #3
    why is a^2 = a + 1? As a field, all these elements need additive inverses do they not? what is the additive inverse for 1?
     
  5. Feb 27, 2015 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    1 is its own additive inverse
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Struggling to understand a field theorm's corollary
Loading...