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  3. 24. show that (x ^p) ? x has p distinct zeros in z

24. Show that (x ^p) ? x has p distinct zeros in Zp, for any prime...

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Advance Math

24. Show that (x ^p) ? x has p distinct zeros in Zp, for anyprime p. Conclude that (x ^p) ? x = x(x ? 1)(x ? 2)· · ·(x ? (p ?1)).

(this is not as simple as showing that each element inZp is a root -- after all, we've

seen that in Z6[x], the polynomialx^2-5x has 4 roots, 0, 5, 2, and 3, but x^2-5 is not equal to(x-0)(x-5)(x-2)(x-3))

Answer & Explanation Solved by verified expert
4.0 Ratings (642 Votes)
Recall Fermats little theorem states that if p is a primenumber then for any integer a Then choose any element then we can find a unique integer be such that the image of a in is Thengives us the fact in the ring rather field Thusthis proves the fact that every element of is    See Answer
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