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
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.
in dummit and foote's abstract algebra the proof is not very clear i guess. he did not define the binary operation between positive integers and members of the field F.A mapping should be defined to make it clear.Also (1+1+1...ntimes).(1+1+....mtimes) can be (1+1+...mn times) clearly due to properties of the field so it is evident that this step answers all the questions asked above,is'nt it??
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