Finiteness of integral closure in a finite separable extension
= Finiteness of integral closure in a finite separable extension
For a normal <Noetherian ring> $A$ that is an <integral domain>, the <integral closure> in a finite <separable field extension> of its <fraction field> is finite as an $A$-<module>. An <integral field basis by denominator clearing> gives a finite lattice. Integral traces place the closure in its finite <trace-dual lattice>; <Noetherianity> makes that submodule finitely generated.