18.1.26. Define an onto homomorphism f: (Z/36Z) [x] →Z/36Z such that ker(ƒ)= (x). (a) Is (x) prime and/or maximal in (Z/36Z)[x]? (b) Let A B = = (6). (3) be an ideal of Z/36Z. Find f¯¹(A). Do the same for (c) Find a familiar ring that is isomorphic to (Z/36Z)[x]/ƒ¯¹(A). Do the same for f¹(A)/(x), and ƒ¯¹(B)/(x). (d) Find a ring of the form (Z/36Z)/?? that is isomorphic to (Z/36Z)[x]/ f-¹(B). (e) Find two maximal ideals in (Z/36Z) [x].
18.1.26. Define an onto homomorphism f: (Z/36Z) [x] →Z/36Z such that ker(ƒ)= (x). (a) Is (x) prime and/or maximal in (Z/36Z)[x]? (b) Let A B = = (6). (3) be an ideal of Z/36Z. Find f¯¹(A). Do the same for (c) Find a familiar ring that is isomorphic to (Z/36Z)[x]/ƒ¯¹(A). Do the same for f¹(A)/(x), and ƒ¯¹(B)/(x). (d) Find a ring of the form (Z/36Z)/?? that is isomorphic to (Z/36Z)[x]/ f-¹(B). (e) Find two maximal ideals in (Z/36Z) [x].
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.6: Algebraic Extensions Of A Field
Problem 7E
Related questions
Question
![18.1.26. Define an onto homomorphism f: (Z/36Z) [x] →Z/36Z such that ker(ƒ)=
(x).
(a) Is (x) prime and/or maximal in (Z/36Z)[x]?
(b) Let A
B =
=
(6).
(3) be an ideal of Z/36Z. Find f¯¹(A). Do the same for
(c) Find a familiar ring that is isomorphic to (Z/36Z)[x]/ƒ¯¹(A). Do
the same for f¹(A)/(x), and ƒ¯¹(B)/(x).
(d) Find a ring of the form (Z/36Z)/?? that is isomorphic to (Z/36Z)[x]/
f-¹(B).
(e) Find two maximal ideals in (Z/36Z) [x].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fbaae5-8d47-4476-8095-8b380294ae7e%2Ffdd8e14f-2668-4301-a1f8-cd52654588a7%2Fzreytn_processed.png&w=3840&q=75)
Transcribed Image Text:18.1.26. Define an onto homomorphism f: (Z/36Z) [x] →Z/36Z such that ker(ƒ)=
(x).
(a) Is (x) prime and/or maximal in (Z/36Z)[x]?
(b) Let A
B =
=
(6).
(3) be an ideal of Z/36Z. Find f¯¹(A). Do the same for
(c) Find a familiar ring that is isomorphic to (Z/36Z)[x]/ƒ¯¹(A). Do
the same for f¹(A)/(x), and ƒ¯¹(B)/(x).
(d) Find a ring of the form (Z/36Z)/?? that is isomorphic to (Z/36Z)[x]/
f-¹(B).
(e) Find two maximal ideals in (Z/36Z) [x].
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