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Chapter 5 Solutions
Elements Of Modern Algebra
- Prove that if R is a field, then R has no nontrivial ideals.arrow_forwardSince 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.arrow_forwardProve that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.arrow_forward
- 15. (See Exercise .) If and with and in , prove that if and only if in . 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 .arrow_forwardProve that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]arrow_forwardLet a0 in the ring of integers . Find b such that ab but (a)=(b).arrow_forward
- 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]arrow_forward15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forwardProve that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.arrow_forward
- 14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.arrow_forwardSuppose 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]arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,