Integral closure of Dedekind domains #
This file shows the integral closure of a Dedekind domain (in particular, the ring of integers
of a number field) is a Dedekind domain.
Implementation notes #
The definitions that involve a field of fractions choose a canonical field of fractions,
but are independent of that choice. The
..._iff lemmas express this independence.
Often, definitions assume that Dedekind domains are not fields. We found it more practical
to add a
(h : ¬ is_field A) assumption whenever this is explicitly needed.
dedekind domain, dedekind ring
We show that an integral closure of a Dedekind domain in a finite separable
field extension is again a Dedekind domain. This implies the ring of integers
of a number field is a Dedekind domain.
Send a set of
x'es in a finite extension
L of the fraction field of
(y : R) • x ∈ integral_closure R L.
L is a finite extension of
K = Frac(A),
L has a basis over
A consisting of integral elements.