Proving Ring Homomorphism of \phi: Zp \rightarrow Zp

[phyguy321]

Homework Statement

Prove that $$\phi$$ : Z_p $$\rightarrow$$ Z_p,
$$\phi$$ (a) = a ^p is a ring homomorphism, find the ker $$\phi$$

Homework Equations

The Attempt at a Solution

So show that a $$_{p}$$ + b $$_{p}$$ = (a + b)^p?
and (ab)^p = (a^p)(b^p)?

 

[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?

 

[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?

 

[Dick]

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.