Algebra Theory?

Question

a. Consider f(x), g(x) E Z(subscript 7)[x], where f(x)=x^7, g(x)=x
Prove that f(c)=g(c) for all c E Z(subscript 7)?

b. Now let f(x), g(x) E Z(subscript 7)[x], where f(x) does not equal g(x) and deg f <7, deg g<7.
Prove that there exists c E Z(subscript 7) such that f(c) does not equal g(c)?

Help would be much appreciated!

Answer 1

a: Three words: Fermat's little theorem

b: Consider the polynomial f-g -- this is also a polynomial of degree less than 7, and because f≠g, f-g is nonzero. Therefore, if we can show that for every nonzero polynomial in Z₇[x] that there is some c∈Z₇ for which it does not evaluate to zero, then there must be some c for which (f-g)(c)≠0, so f(c)≠g(c). So let us suppose that a nonzero polynomial p is identically 0 over Z₇ -- then ∀n∈Z₇, (x-n) | p, so the polynomial x (x-1) (x-2) (x-3) (x-4) (x-5) (x-6) ≡ x⁷ - x must divide p, so since p is nonzero, p must have degree greater than or equal to 7. Therefore, for any nonzero polynomial p of degree less than 7, there must be some c∈Z₇ such that p(c)≢0, so the theorem is proved. Q.E.D.

Answer 2

Although this is more of calculus than algebra, that other guy is rite on. I couldn't explain it any better or easier. lol