Let F be a field and let f(x) be a 5 points polynomial in F[x] that is reducible over F. Then * O < f(x) > is a maximal in F[x] None of the choices < f(x) > is not a prime ideal in F[x] < f(x) > is a prime ideal but not maximal in F(x]
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![Let F be a field and let f(x) be a
5 points
polynomial in F[x] that is reducible
over F. Then *
O < f(x) > is a maximal in F[x]
None of the choices
< f(x) > is not a prime ideal in F[x]
< f(x)
> is a prime ideal but not maximal in
F(x]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd0b627c-ed12-4d47-825c-84b9bf5b3a12%2Fc73079eb-482a-4e19-b00b-138a6f93c3b4%2Fb0wfn5n_processed.png&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 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.Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
- 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.Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.
- 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]Let Q denote the field of rational numbers, R the field of real numbers, and C the field of complex. Determine whether each of the following polynomials is irreducible over each of the indicated fields, and state all the zeroes in each of the fields. a. x22 over Q, R, and C b. x2+1 over Q, R, and C c. x2+x2 over Q, R, and C d. x2+2x+2 over Q, R, and C e. x2+x+2 over Z3, Z5, and Z7 f. x2+2x+2 over Z3, Z5, and Z7 g. x3x2+2x+2 over Z3, Z5, and Z7 h. x4+2x2+1 over Z3, Z5, and Z7In 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 . , ,