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- 8. Prove that the characteristic of a field is either 0 or a prime.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 .)Use 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 .
- True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field of .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 if R is a field, then R has no nontrivial ideals.
- 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]Prove Theorem If and are relatively prime polynomials over the field and if in , then in .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]
- Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over overIf 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.