Unformatted text preview: L D [f M W K ± M ± L and M is ﬁnite–dimensional g . If there is an N 2 N such that dim K .M/ ² N whenever K ± M ± L and M is ﬁnite–dimensional, then also dim K .L/ ² N . 9.5.2. This exercise gives the proof of Proposition 9.5.6 . We suppose that K ± L is a ﬁnite–dimensional ﬁeld extension, that A;B are ﬁelds 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 KŒxŁ be the minimal polynomial for ˛ . Then B is a splitting ﬁeld for p.x/ over K , and the roots of p.x/ are distinct, by Theorem 9.4.14 . (a) Show that A ³ B is Galois over A . Hint: A ³ B D A.˛/ ; show that A ³ B is a splitting ﬁeld for p.x/ 2 AŒxŁ ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08