27 Thus, our rational map is given by T 1 7→ T 2-1 T 2 + 1 , T 2 7→-2 T T 2 + 1 . Next note that the obtained map is birational. The inverse map is given by T 7→ T 2 T 1-1 . In particular, we see that R ( V ( T 2 1 + T 2 2-1)) ∼ = k ( T 1 ) . The next theorem, although sounding as a deep result, is rather useless for concrete applications. Theorem 4.10. Assume k is of characteristic0 . Then any irreducible aﬃne algebraic k-set is birationally isomorphic to an irreducible hypersurface. Proof. Since R ( V ) is a ﬁnitely generated ﬁeld over k , it can be obtained as an algebraic extension of a purely transcendental extension L = k ( t 1 , . . . , t n ) of k . Since char ( k ) = 0 , R ( V ) is a separable extension of L , and the theorem on a primitive element applies (M. Artin, ”Algebra”, Chapter 14, Theorem 4.1): an algebraic extension K/L of characteristic zero is generated by one element x ∈ K . Let k [ T 1 , . . . , T n +1 ] → R ( V ) be deﬁned by sending T i to
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This note was uploaded on 01/08/2012 for the course MATH 299 taught by Professor Wei during the Spring '09 term at SUNY Stony Brook.