Ring homomorphism

1. Oct 22, 2008

phyguy321

1. The problem statement, all variables and given/known data
Prove that $$\phi$$ : Zp $$\rightarrow$$ Zp,
$$\phi$$ (a) = a p is a ring homomorphism, find the ker $$\phi$$

2. Relevant equations

3. The attempt at a solution
So show that a $$_{p}$$ + b $$_{p}$$ = (a + b)p?
and (ab)p = (ap)(bp)?

Last edited: Oct 22, 2008
2. Oct 22, 2008

Dick

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. Oct 23, 2008

phyguy321

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. Oct 23, 2008

Dick

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.