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Question on Galois

  1. Dec 4, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove that if p is a prime then [tex](\mathbb{Z}/p^2)^x \cong C_p[/tex] [tex]\times[/tex] [tex]C_{p-1} \cong C_{p(p-1)}[/tex] where [tex]C_n[/tex] denotes the cyclic group of order n.



    Next part is to describe the Galois structure of the field of [tex]p^2[/tex] roots of unity, p is prime for the cases of a. p=5, b. p=7. How many intermediate fields are there in these cases?


    2. Relevant equations



    3. The attempt at a solution
    My idea was to consider a homomorphism [tex]\mathbb{Z}/p^2 \rightarrow \mathbb{Z}/p [/tex] and observe that [tex](\mathbb{Z}/p)^\times[/tex] is a cyclic group of order p-1. Then maybe use proof by induction that if p is an odd prime then [tex](\mathbb{Z}/p^n)^\times[/tex] is cyclic for all n.
     
  2. jcsd
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