17. If E is an extension of F and f (x) e F [x] and if is an automorphism of E leaving every element of F fixed, prove that o must take a root of f (x) in E into a root of f (x) in E.
17. If E is an extension of F and f (x) e F [x] and if is an automorphism of E leaving every element of F fixed, prove that o must take a root of f (x) in E into a root of f (x) in E.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![17. If E is an extension of F and f (x) e F [x] and if o is
an automorphism of E leaving every element of F fixed, prove that o must
take a root of f (x) in E into a root off(x) in E.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b53687f-318e-45de-bb66-94da174d66b6%2Fb57d5e2e-4f40-41b7-ac62-d3707491ffc0%2Fbv9qlsh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:17. If E is an extension of F and f (x) e F [x] and if o is
an automorphism of E leaving every element of F fixed, prove that o must
take a root of f (x) in E into a root off(x) in E.
Transcribed Image Text:Let R be a commutative ring and let A be an ideal
of R. Show that
VA = {x e R: x" e A for some positive integer n}
%3D
is an ideal of R such that
(i) AS VA
(iii) If R has unity and VA = R, then A = R.
(ii) VA = VA
%3D
%3D
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