Let F be a field of 4 elements and let f(X) e F[X] be an irreducible polynomial of degree 4. How many elements has the field F(X]/(F(X] S(X))?
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![Let F be a field of 4 elements and let f(X) e F(X] be an irreducible polynomial of degree 4. How many elements has
the field F[X]/(F[X] · $(X)) ?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd01d0b11-583b-4188-99cd-5c5c99346052%2F1590a8b6-c12f-404a-b2a3-b7fabac4dd9a%2Fvpadfeik_processed.jpeg&w=3840&q=75)
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- Prove Theorem If and are relatively prime polynomials over the field and if in , then 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.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 .
- 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]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 .)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 .
- Let 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.Let where is a field and let . Prove that if is irreducible over , then is irreducible over .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 . , ,