Let a EC be a primitive nth root of unity.Show that GalQ(Q(a)) is isomorphic to thegroup En = {kEN |1 ≤ k ≤n,gcd (k,n) = 1}.

Answered: Image /qna-images/answer/146e1df7-bd56-4cf1-8dc6-61bcc5f0cd95.jpg

Answered: Let a EC be a primitive nth root of…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 5E

See similar textbooks

Similar questions

31. Prove statement of Theorem : for all integers and .
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
Show that the ideal is a maximal ideal of .
3. Let be an integral domain with positive characteristic. Prove that all nonzero elements of have the same additive order .
Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
14. Let be an ideal in a ring with unity . Prove that if then .
9. If denotes the unity element in an integral domain prove that for all .
29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .
34. If is an ideal of prove that the set is an ideal of . The set is called the annihilator of the ideal . Note the difference between and (of Exercise 24), where is the annihilator of an ideal and is the annihilator of an element of.
6. Prove that if is any element of an ordered integral domain then there exists an element such that . (Thus has no greatest element, and no finite integral domain can be an ordered integral domain.)

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS