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## Homework Statement

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.