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Abstract Linear Algebra

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Running head: Abstract linear algebra 1 Abstract linear algebra Name of student Name of professor Name of course Name of institution 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 over F Since F’=g(x)=xm F(x-1); It follows that g(x) is also a polynomial over F of degree M. (b) Show that α-1 is a root of g(x). If f and g are relatively prime, their greatest common divisor is 1, so there are polynomials a(x) and b(x) over f such that a(x)f(x)+b(x)g(x)=1. If α is a common root of f and g, then substitute α for x; a(α)f(x)+b(α)g(α)=0 which shows that the greatest common divisor d(x) of both f(x) and g(x) is constant. Let K be an extension of f in which d(x) has a root α. Since d(x) divides both f(x) and g(x), α is a r ...
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