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.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.1: Ideals And Quotient Rings
Problem 24E: 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set...
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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.
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.
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
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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