Algebraic structures and Abstract Linear Algebra Questions

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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 fis 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*.)

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