List all the subfields of the field F262144 in which the polynomial x³ + x² + 1 has a root.
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Q: 1 (2¹) 4 4
A: Given matrix as, A=-2144
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- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.8. Prove that the characteristic of a field is either 0 or a prime.
- Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inUse Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
- Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .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.
- Prove that any ordered field must contain a subfield that is isomorphic to the field of rational numbers.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 .)Factors each of the polynomial in Exercise 1316 as a product of its leading coefficient and a finite number of monic irreducible polynomial over the field of rational numbers. 6x4+x3+3x214x8
