Question

Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this to show that you can write απ¡n terms of (c) Suppose we have two such expressions that are equal β = X bial = Xcia' .Prove that the polynomial g(x) defined by g(x) = i-0 i-0 (b, - ci)r must be the zero polynomial Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every element β E E can be expressed uniquely in the form β = bo + bla + . . . + bn-jan-1 for some bi E F. (a) Show that every element can be written as f(a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ao. Ụse this to show that you can write o" in terms of (c) Suppose we have two such expressions that are equal i-0 i-0 Prove that the polynomial g(x)-Σ(bi-ci)ri must be the zero polynomial

Answer 1

m-2 1-2 n1