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 coefﬁcients in K . Corollary 7.3.11. Let K ² L be a ﬁeld extension. The set of elements of L that are algebraic over K form a subﬁeld of L . In particular, the set of algebraic numbers (complex numbers that are algebraic over Q ) is a countable ﬁeld. Proof. Let A denote the set of elements of L that are algebraic over K . It sufﬁces to show that A is closed under the ﬁeld 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 sufﬁces that K.a;b/ ² A . But this follows from Proposition 7.3.9 ....
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- Fall '08
- Algebra, Proposition, Complex number, Ki Œai C1, Ki C1, C1 D Ki