The Nagata automorphism is wild.

Shestakov, Ivan P; Umirbaev, Ualbai U · Proc Natl Acad Sci U S A · 2003

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Abstract

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms.