Given f(x) = 9+8x² + x¹, find the following: a. The galois group of f(x). b. The subfields of splitting field F of f(x) over base field Q.
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- Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)
- Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.In Exercises , a field , a polynomial over , and an element of the field obtained by adjoining a zero of to are given. In each case: Verify that is irreducible over . Write out a formula for the product of two arbitrary elements and of . Find the multiplicative inverse of the given element of . , ,Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.