If f is a field containing an in finle number of destinetof distinet elements the mapping frof is an isomovphism.of the algebra, of polyhomials over f onto the alabra ofPolynonial funcdions over. f

Answered: Image /qna-images/answer/61c87ac7-5520-4b48-b915-b43c39f9e053.jpg

Answered: If f is a field containing an in finte…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 16E: Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the...

See similar textbooks

Similar questions

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.
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+.
If F is an ordered field, prove that F contains a subring that is isomorphic to . (Hint: See Theorem 5.35 and its proof.) Theorem 5.35: Isomorphic Images of If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then, D is isomorphic to the ring of all integers.
Prove that any ordered field must contain a subfield that is isomorphic to the field of rational numbers.
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]
8. Prove that the characteristic of a field is either 0 or a prime.
Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
Prove that if R is a field, then R has no nontrivial ideals.
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
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.)
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]
21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

If f is a field containing an in finte humber of destinct of distinst elen