Let F be a field of characteristic not equal to 2. Let D1, D₂ € F, neither of which is a square in F. Let K = F = F/VR₁ VD.
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- Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].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 where is a field and let . Prove that if is irreducible over , then is irreducible over .
- 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][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]
- 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]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 . , ,If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]
- 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.For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]