Full solution to the question.

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The question will be checked deeply with follow-up questions (if needed) in order to make sure the solution is correct, please don’t waste each others time.

Let F be a finite field, and f(x) be a polynomial over F of degree m, with

nonzero constant term. Let a be a root of fin some extension field K of F.

Define the reciprocal of f to be: g(x)=x” f(x’)

(a) Show that g(x) is a polynomial over F of degree m.

(b) Show that a’ is a root of g(x).

(c) Show that f is irreducible if and only if g is irreducible.

(d)

Show that a is a generator for K* if and only if a’ is a generator for

K* (In this case we say that f is a primitive polynomial for K*.)