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- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.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]Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][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]Find all monic irreducible polynomials of degree 2 over Z3.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 R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .Prove that any ordered field must contain a subfield that is isomorphic to the field of rational numbers.