2. We showed in class that if R is a ring a characteristic p (where p is prime) that (a+b)^p = a^p + b^p. (All of the binomial coefficients in the middle of the expansion are divisible by p and thus 0 in R.) Given this fact, prove the following: (a) Let F be a finite field of characteristic p. Show that ?(x) = x^p is a ring homomorphism. (b) Show that ? is bijective and thus an automorphism. (Hint: show that ? is injective and thus bijective.) (c) Show that ? is trivial on Z_p, but is non-trivial on GF(p^n) for n > 1.
Added by Raul M.
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First, let's show that o(x) preserves addition. Let a, b be elements in F. Then we have: o(a + b) = (a + b)P Expanding this using the binomial theorem, we get: (a + b)P = aP + bP + (binomial coefficients divisible by p) Since all the binomial coefficients in Show more…
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