Let K be a field extension field F and let a € K be algebric over F.
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- 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.Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].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 .)Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.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]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 .6. Prove that if is a permutation on , then is a permutation on .
- Prove that if f is a permutation on A, then (f1)1=f.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 a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.