Cancellation Problems and Dimension Theory

What is it about?

The authors assert the following interesting results. (1) Let $A_1$ and $A_2$ be affine domains (finitely generated integral domains) over an arbitrary field $k$. Suppose that a polynomial ring $A_1[t]$ is embedded into a polynomial ring $A_2[t]$ as $k$-algebras. Then $A_1$ is embedded into $A_2$ as $k$-algebras. (2) Let $K_1$ and $K_2$ be affine fields (finitely generated fields) over an arbitrary field $k$. Suppose that a simple transcendental extension $K_1(t)$ is embedded into $K_2(t)$. Then $K_1$ is embedded into $K_2$. The authors also discuss generalizations of Lüroth's theorem, Zariski cancellation problems and other related topics. Reviewed by M. Miyanishi

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