Algebraic structures and Abstract Linear Algebra Questions

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Let F be finite field, and f(X) be a polinomial over F of degree m, with nonzero constant term.

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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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Running head: Abstract linear algebra

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Abstract linear algebra
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Abstract linear algebra
(a) Show that g(x) is a polynomial over F of degree M.
Let F be a field, and let f(x) be a nonzero polynomial over F of degree M.
Hence reciprocal of F, F’is also a nonzero polynomial ov...


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