Check that the union of a countable collection of o-fields Bj, j 2 1 need not be a o-field even if B; C B;+1. Is a countable union of o-fields whether monotone or not a field? Hint: Try setting 2 equal to the set of positive integers and let C; be all subsets of {1, ..., j} and B; = o (Cj). If Bi, i = 1,2 are two o-fields, B1 U B2 need not be a field. %3D

Check that the union of a countable collection of o-fields Bj, j 2 1 need not be a o-field even if B; C B;+1. Is a countable union of o-fields whether monotone or not a field? Hint: Try setting 2 equal to the set of positive integers and let C; be all subsets of {1, ..., j} and B; = o (Cj). If Bi, i = 1,2 are two o-fields, B1 U B2 need not be a field. %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 22E: Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e...

See similar textbooks

Similar questions

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]
Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [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]
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 .
Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
Prove that if R is a field, then R has no nontrivial ideals.
8. Prove that the characteristic of a field is either 0 or a prime.
14. Prove or disprove that is a field if is a field.
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.
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 .
Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
Prove 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+.