a polynomial of deg. F. then there exists of F in which f(x) h
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![Let f (x) be a polynomial of degree n21
over a field F. then there exists a finite
extension K of F in which f(x) has n roots.
such that [K:F]s n!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb5fe9ac-dc42-43ea-89ff-7259eb0926e0%2Fcae8ad3a-3f58-42ae-9d5a-8df4455a98b5%2F0cfqkfq_processed.jpeg&w=3840&q=75)
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- 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.Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inProve Theorem If and are relatively prime polynomials over the field and if in , then 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 .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 .8. Prove that the characteristic of a field is either 0 or a prime.
- Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .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 . , ,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 .
- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in1. Find a monic polynomial of least degree over that has the given numbers as zeros, and a monic polynomial of least degree with real coefficients that has the given numbers as zeros. a. b. c. d. e. f. g. and h. andLet ab in a field F. Show that x+a and x+b are relatively prime in F[x].