algebraic over k(α₁, ₂,..., α). Therefore, [k(α₁, ₂,..., an) (an+1): k(α₁, ₂,..., α₁)] is finite. By the induction hypothesis, [k(a₁, 0₂,..., an): k] is finite. Thus [F:k] [k(α₁, ₂,..., anan): k] = [k(α₁, ₂,..., α₁)(a): k(α₁, α₂,..., a)][k (α₁, ₂,..., α):k] is finite. Hence, F/k is a finite extension for any F generated by a finite number of U
algebraic over k(α₁, ₂,..., α). Therefore, [k(α₁, ₂,..., an) (an+1): k(α₁, ₂,..., α₁)] is finite. By the induction hypothesis, [k(a₁, 0₂,..., an): k] is finite. Thus [F:k] [k(α₁, ₂,..., anan): k] = [k(α₁, ₂,..., α₁)(a): k(α₁, α₂,..., a)][k (α₁, ₂,..., α):k] is finite. Hence, F/k is a finite extension for any F generated by a finite number of U
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 11E
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