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Unformatted text preview: L D [f M W K M L and M is nitedimensional g . If there is an N 2 N such that dim K .M/ N whenever K M L and M is nitedimensional, then also dim K .L/ N . 9.5.2. This exercise gives the proof of Proposition 9.5.6 . We suppose that K L is a nitedimensional eld extension, that A;B are elds interme-diate between K and L , and that B is Galois over K . Let be an element of B such that B D K./ (Proposition 9.5.1 ). Let p.x/ 2 Kx be the minimal polynomial for . Then B is a splitting eld for p.x/ over K , and the roots of p.x/ are distinct, by Theorem 9.4.14 . (a) Show that A B is Galois over A . Hint: A B D A./ ; show that A B is a splitting eld for p.x/ 2 Ax ....
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- Fall '08