[1] If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).
); in this context, the integral elements are usually called algebraic integers.
The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory.
There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the ring of integers for an algebraic field extension
can be found by constructing the minimal polynomial of an arbitrary element
and finding number-theoretic criterion for the polynomial to have integral coefficients.
Given an element b in B, the following conditions are equivalent: The usual proof of this uses the following variant of the Cayley–Hamilton theorem on determinants: This theorem (with I = A and u multiplication by b) gives (iv) ⇒ (i) and the rest is easy.
This situation is applicable in algebraic number theory when relating the ring of integers and a field extension.
is a union (equivalently an inductive limit) of subrings that are finitely generated
is noetherian, transitivity of integrality can be weakened to the statement: Finally, the assumption that
Let B be a ring that is integral over a subring A and k an algebraically closed field.
Let B be a ring and A a subring that is a noetherian integrally closed domain (i.e.,
This is called the Splitting of prime ideals in Galois extensions.
is a purely inseparable extension (need not be normal), then
Specifically, for a multiplicatively closed subset S of A, the localization S−1A' is the integral closure of S−1A in S−1B, and
If A is a subring of a field K, then the integral closure of A in K is the intersection of all valuation rings of K containing A.
The notion of integral closure of an ideal is used in some proofs of the going-down theorem.
Because the notion has origin in algebraic number theory, the conductor is denoted by
[21] If S is a multiplicatively closed subset of A, then If B is a subring of the total ring of fractions of A, then we may identify Example: Let k be a field and let
The integral closure of a Dedekind domain in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring.
[citation needed] Let A be a noetherian integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure
[24] This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form).
Let A be a finitely generated algebra over a field k that is an integral domain with field of fractions K. If L is a finite extension of K, then the integral closure
It is clear that it is enough to show the assertion when L/K is either separable or purely inseparable.
The separable case is noted above, so assume L/K is purely inseparable.
By the normalization lemma, A is integral over the polynomial ring
[26] More precisely, for a local noetherian ring R, we have the following chains of implications:[27] Noether's normalisation lemma is a theorem in commutative algebra.
Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are algebraically independent over K such that A is finite (and hence integral) over B = K[y1,..., ym].
The class of integral morphisms is more general than the class of finite morphisms because there are integral extensions that are not finite, such as, in many cases, the algebraic closure of a field over the field.
Let A be an integral domain and L (some) algebraic closure of the field of fractions of A.