A NOTE ON THE BIRATIONAL CANCELLATION PROBLEM

[摘要] Let k be any field, K1 and K2 finitely generated extension fields of k, K1(x1, ..., x(n)) and K2(y1, ..., y(n)) the rational function fields of n variables over K1 and K2, respectively. Suppose that sigma: K1(x1, ..., x(n)) --> K2(y1, ..., y(n)) is a k-isomorphism of K1(x1, ..., x(n)) onto K2(y1, ..., y(n)). Theorem A. If sigma(x(i) = y(i) for 1 less-than-or-equal-to i less-than-or-equal-to n, k is infinite and Aut(k)(K1) is finite, then sigma(K1) = K2. Theorem B. (1) If K1 has no proper k-endomorphism, then K1 is k-isomorphic to K2. (2) If Aut(k)(K1) is finite and K1 has no proper k-endomorphism, then sigma(K1) = K2.

[发布日期] 1992-02-28