(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Assuming the mapping Z --> F defined by n --> n * 1F = 1F + ... + 1F (n times) is a ring homomorphism, show that its kernel is of the form pZ, for some prime number p. Therefore infer that F contains a copy of the finite field Z/pZ.

Also prove now that F is a finite dim vector space over Z/pZ; if this dim. is denoted d, then show that F has exactly p^d elements.

I know that the kernel of a ring homomorphism is defined as ker(f) = {a in Z : f(a) = 0}

but I am still having trouble exactly where to go from this...it appears that the only element of Z s.t. f(a) = 0, is 0 which would map to 0 * 1F = 0. But how is this of the form pZ, for some prime p??

Any help or push in the right direction would be great...thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Finite Fields and ring homomorphisms HELP!

**Physics Forums | Science Articles, Homework Help, Discussion**