1. In this exercise we will prove that Z[√-5] is an integral domain but not aunique factorisation domain (UFD).Show that for any integer m, the setZ√√m] = {a+b√√m: a, b = Z}is a subring of the field C. Use this to conclude that Z[√√m] is an integraldomain.

Detailed explanation:Step 1: Show Z[√(−5)] is a subring of CA subring of a field is a subset that…

Answered: 1. In this exercise we will prove that…

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

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]
Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
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]
8. Prove that the characteristic of a field is either 0 or a prime.
Prove that if R is a field, then R has no nontrivial ideals.
14. Prove or disprove that is a field if is a field.
Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .
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 .)
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
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.)

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS

1. In this exercise we will prove that Z[√-5] is an integral domain but not a unique factorisation domain (UFD). Show that for any integer m, the set Z√√m] = {a+b√√m: a, b = Z} is a subring of the field C. Use this to conclude that Z[√√m] is an integral domain.