Algebraic Extensions
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Question 1
If [K:F] is finite and u is algebraic over K ,prove that [F(u):F] divides [K(u):F]
Hint:[F(u):F] and [K(u):F(u)] are finite by Theorems 10.4,10.7 and 10.9
Apply Theorem 10.4 to
Theorem 10.4
Let F,K and L be fields with F ⊆ K ⊆ L .If [K:F] and [L:K] are finite ,then L is a finite dimensional extension of F and [L:F] = [L:K] [K:F]
Theorem 10.7
Let K be an extension field of F and u an algebraic element over F with minimal polynomial p(x) of degree n.Then
(1) F(u) F[x]/(p(x))
(2) { ,u , , ................., } is a basis of the vector space F(u) over F
(3) [F(u):F] = n
Theorem 10.7 shows that when u is algebraic over F,then F(u) does not depend on K but is completely determined by F[x] and the minimal polynomial p(x).Consequently ,we sometimes say that F(u) is the field obtained by adjoining u to F
Theorem 10.9
If K is a finite-dimensional extension field of F,then K is an algebraic extension of F.
Question 2
Assume that u,v K are algebraic over F,with minimal polynomial p(x) and q(x),respectively.
(a) If deg p(x) = m and deg q(x) = n and (m,n) = 1, prove that [F(u,v):F] = mn.
(b) Show by example that the conclusion of part (a) may be false if m and n are not relatively prime.
(c) What is [Q( ):Q]?
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Solution Summary
When working with algebraic extensions, it is often useful to introduce intermediate extensions. This solution addresses one of the fundamental results that enables us to do this, and then illustrates some applications. The solution comprises a 1 page document written in Word with equations written in Mathtype.
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