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Unformatted text preview: Furthermore, if L D K.a 1 ;:::;a n / , where the a i are algebraic over K , then L consists of polynomials in the a i with coefcients in K . Corollary 7.3.11. Let K L be a eld extension. The set of elements of L that are algebraic over K form a subeld of L . In particular, the set of algebraic numbers (complex numbers that are algebraic over Q ) is a countable eld. Proof. Let A denote the set of elements of L that are algebraic over K . It sufces to show that A is closed under the eld operations; that is, for all a;b 2 A , the elements a C b , ab , a , and b 1 (when b ) also are elements of A . For this, it certainly sufces that K.a;b/ A . But this follows from Proposition 7.3.9 ....
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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