Let F be a field of characteristic not equal to 2. Let a, b E F such that b is not T CL If 1
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- [Type here] True or False Label each of the following statements as either true or false. 3. Every integral domain is a field. [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 that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
- Label each of the following as either true or false. If a set S is not an integral domain, then S is not a field. [Type here][Type here]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.
- 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 a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros 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 .
- Prove that if R is a field, then R has no nontrivial ideals.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]