Ring homomorphism

  • Thread starter phyguy321
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  • #1
phyguy321
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Homework Statement


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]


Homework Equations





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:

Answers and Replies

  • #2
Dick
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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?
 
  • #3
phyguy321
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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?
 
  • #4
Dick
Science Advisor
Homework Helper
26,263
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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?

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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