Let F be a field and let I = {a„x" + a„-1.X"-1 a, + an-1 + · ··+ ao = 0}. ...+ ao I an, an-1, . .. , ao E F and Show that I is an ideal of F[x] and find a generator for I.
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![Let F be a field and let
I = {a„x" + a„-1.X"-1
a, + an-1 + · ··+ ao = 0}.
...+ ao I an, an-1, . .. , ao E F and
Show that I is an ideal of F[x] and find a generator for I.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c401a4d-5531-4107-bc93-6031cf60f754%2Fd0ef3f39-d4d0-42ec-a256-8e5044a498d8%2Fo6x09m.jpeg&w=3840&q=75)
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- 8. Prove that the characteristic of a field is either 0 or a prime.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.Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]
- 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 .Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
- 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 Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inLet F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.
- 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 .)Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inTrue 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 .