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## Main Question or Discussion Point

Why is the characteristic of a finite field a prime number???!

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Why is the characteristic of a finite field a prime number???!

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A finite field clearly has a characteristic (among the elements 1, 1 + 1, 1 + 1 + 1, ... there must be two that equal one another, since we have only finitely many elements in the field). Let p be the least number of ones we need to add up in order to get 0. Suppose p = nm with 1 < n, m < p (i.e. p is not prime). Then

0 = 1 + 1 ... + 1 (p times) = p = nm = (1 + ... + 1)(1 + ... + 1) := ab

where a is the first paranthesis (containing n ones) and b is the second paranthesis (containing m ones). But since we're in a field, this implies that either a or b is 0, contradicting the fact that p minimal.

0 = 1 + 1 ... + 1 (p times) = p = nm = (1 + ... + 1)(1 + ... + 1) := ab

where a is the first paranthesis (containing n ones) and b is the second paranthesis (containing m ones). But since we're in a field, this implies that either a or b is 0, contradicting the fact that p minimal.

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Hurkyl

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mathwonk

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Would you mind expanding on this explanation a bit? What is the significance of a nontrivial zero-divisor? Thanks!

morphism

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How can a field have a nontrivial zero-divisor?Would you mind expanding on this explanation a bit? What is the significance of a nontrivial zero-divisor? Thanks!

mathwonk

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mathwonk

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(the point is that in a field if AB=0 then either A=0 or B=0.)

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