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Ring homomorphism

  1. Oct 22, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that [tex]\phi[/tex] : Zp [tex]\rightarrow[/tex] Zp,
    [tex]\phi[/tex] (a) = a p is a ring homomorphism, find the ker [tex]\phi[/tex]

    2. Relevant equations

    3. The attempt at a solution
    So show that a [tex]_{p}[/tex] + b [tex]_{p}[/tex] = (a + b)p?
    and (ab)p = (ap)(bp)?
    Last edited: Oct 22, 2008
  2. jcsd
  3. Oct 22, 2008 #2


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    Yes, show those equalities mod p. The second is easy. For the first think about the binomial theorem. Is p supposed to be a prime?
  4. Oct 23, 2008 #3
    p is prime.
    so show a mod p + b mod p = (a+b) mod p
    and (ab) mod p = (a mod p)*(b mod p)
    how do i do that?
  5. Oct 23, 2008 #4


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    Those are always true whether p is prime or not. You must have already proved them. Your problem is to prove (a+b)^b mod p=(a+b) mod p when p is prime. I told you how to do that. Use the binomial theorem on (a+b)^p.
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