26LECTURE 4. IRREDUCIBLE ALGEBRAIC SETS AND RATIONAL FUNCTIONSWe haveP(T1, . . . , Tn)≡0onV\Z.SinceV\Zis dense in the Zariskitopology,P≡0onV, i.e.P∈I(V). This shows that under the mapR(W)→R(V),Fgoes to0. Since the homomorphismR(W)→R(V)is injective (anyhomomorphism of fields is injective) this is absurd.In particular, takingW=A1k(K), we obtain the interpretation of elementsof the fieldR(V)as non-constant rational functionsV-→Kdefined on anopen subset ofV(the complement of the set of the zeroes of the denominator).From this point of view, the homomorphismR(W)→R(V)defining a rationalmapf:V- →Wcan be interpreted as the homomorphismf*defined by thecompositionφ→φ◦f.Definition 4.4.A rational mapf:V- →Wis calledbirationalif the cor-responding field homomorphismf*:R(W)→R(V)is an isomorphism. Twoirreducible affine algebraic setsVandWare said to bebirationally isomorphicif there exists a birational map fromVtoW.
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