Nagata's compactification theorem shows that
algebraic varieties can be embedded in
complete varieties. The
Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a
group. In 1959, he introduced a
counterexample to the general case of
Hilbert's fourteenth problem on
invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative
Noetherian ring that is not
catenary, and a commutative Noetherian ring of infinite
dimension.
Nagata's conjecture on curves concerns the minimum
degree of a
plane curve specified to have given multiplicities at given points; see also
Seshadri constant.
Nagata's conjecture on automorphisms concerns the existence of wild
automorphisms of
polynomial algebras in three variables. Recent work has solved this latter problem in the affirmative. ==Selected works==