INTRODUCTION TO ALGEBRAIC GEOMETRY-page31

INTRODUCTION TO ALGEBRAIC GEOMETRY-page31 - 27 Thus, our...

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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 characteristic 0 . 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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