College Algebra Exam Review 410

# College Algebra Exam Review 410 - K ± M ± L are ﬁeld...

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Chapter 9 Field Extensions – Second Look This chapter contains a systematic introduction to Galois’s theory of ﬁeld extensions and symmetry. The ﬁelds in this chapter are general, not nec- essarily subﬁelds of the complex numbers nor even of characteristic 0. We will restrict ours attention, however, to so called separable algebraic ex- tensions. 9.1. Finite and Algebraic Extensions In this section, we continue the exploration of ﬁnite and algebraic ﬁeld extensions, which was begun in Section 7.3 . Recall that a ﬁeld extension K ± L is said to be ﬁnite if dim K .L/ is ﬁnite and is called algebraic in case each element of L satisﬁes a polynomial equation with coefﬁcients in K . Proposition 9.1.1. Suppose that
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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.

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