Unformatted text preview: K ± M ± L are ﬁeld extensions, (a) If M is algebraic over K and b 2 L is algebraic over M , then b is algebraic over K . (b) If M is algebraic over K , and L is algebraic over M , then L is algebraic over K . Proof. For part (a), since b is algebraic over L , there is a polynomial p.x/ D a C a 1 x C ²²² C a n x n with coefﬁcients in M such that p.b/ D . But this implies that b is algebraic over K.a ;:::;a n / , and, therefore, K.a ;:::;a n / ± K.a ;:::;a n /.b/ D K.a ;:::;a n ;b/ is a ﬁnite ﬁeld extension, by Proposition 7.3.6 . Since M is algebraic over K , the a i are algebraic over K , and, therefore, K ± K.a ;:::;a n / is a 420...
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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
 EVERAGE
 Algebra, Complex Numbers

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