1. Let F be a field. Suppose that the polynomial ƒ(x) = F[x] is a unit in the polynomial ring F[x]. Prove that f(x) = a must be a constant polynomial, where a E F is a non-zero element of the field. Hint: Let g(x) = F[x] be the multiplicative inverse of f(x). What can you say about deg f(x) and deg g(x)?
1. Let F be a field. Suppose that the polynomial ƒ(x) = F[x] is a unit in the polynomial ring F[x]. Prove that f(x) = a must be a constant polynomial, where a E F is a non-zero element of the field. Hint: Let g(x) = F[x] be the multiplicative inverse of f(x). What can you say about deg f(x) and deg g(x)?
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 13E
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