Abstract Algebra: Algebraic Extensions

Please see the attachment to see these questions properly.

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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]?

Attached file(s):

abstract2.docx  View File