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10 Nov 2019
Let a be a fixed element of a field F and define a map eva FrF by eva(f(x)) f(a). (a) Prove that eva is a surjective ring homomorphism. We call eva an evaluation (b) For an element a ? F, let Ka-{g(z) ? Flr] I eta (g(z))-OF). Prove that Ka (c) Describe what it means for a polynomial h(x) to be in Ka in terms of roots of homomorphism is an ideal in F12]. polynomials. Show that the ideal Ka is a principal ideal. What is it generated by?
Let a be a fixed element of a field F and define a map eva FrF by eva(f(x)) f(a). (a) Prove that eva is a surjective ring homomorphism. We call eva an evaluation (b) For an element a ? F, let Ka-{g(z) ? Flr] I eta (g(z))-OF). Prove that Ka (c) Describe what it means for a polynomial h(x) to be in Ka in terms of roots of homomorphism is an ideal in F12]. polynomials. Show that the ideal Ka is a principal ideal. What is it generated by?
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