Integral element

In commutative algebra, an element "b" of a commutative ring "B" is said to be integral over "A", a subring of "B", if there are "n" ≥ 1 and "a""j" in "A" such that
That is to say, "b" is a root of a monic polynomial over "A". The set of elements of "B" that are integral over "A" is called the integral closure of "A" in "B". It is a subring of "B" containing "A". If every element of "B" is integral over "A", then we say that "B" is integral over "A", or equivalently "B" is an integral extension of "A".
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).
The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., formula_2 or formula_3); 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.
In this article, the term "ring" will be understood to mean "commutative ring" with a multiplicative identity.
Examples.
Integral closure 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 formula_4 (or formula_5).
Integral closure of integers in rationals.
Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
Quadratic extensions.
The Gaussian integers, are the complex numbers of the form formula_6, and are integral over Z. formula_7 is then the integral closure of Z in formula_8. Typically this ring is denoted formula_9.
The integral closure of Z in formula_10 is the ringformula_11this example and the previous one are examples of quadratic integers. The integral closure of a quadratic extension formula_12 can be found by constructing the minimal polynomial of an arbitrary element formula_13 and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the quadratic extensions article.
Roots of unity.
Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z[ζ]. This can be found by using the minimal polynomial and using Eisenstein's criterion.
Ring of algebraic integers.
The integral closure of Z in the field of complex numbers C, or the algebraic closure formula_14 is called the "ring of algebraic integers".
Other.
The roots of unity, nilpotent elements and idempotent elements in any ring are integral over Z.
Integral closure in geometry.
In geometry, integral closure is closely related with normalization and normal schemes. It is the first step in resolution of singularities since it gives a process for resolving singularities of codimension 1.
Equivalent definitions.
Let "B" be a ring, and let "A" be a subring of "B". 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. Coincidentally, Nakayama's lemma is also an immediate consequence of this theorem.
Elementary properties.
Integral closure forms a ring.
It follows from the above four equivalent statements that the set of elements of formula_28 that are integral over formula_29 forms a subring of "formula_28" containing formula_29. (Proof: If "x", "y" are elements of "formula_28" that are integral over formula_29, then formula_34 are integral over formula_29 since they stabilize formula_36, which is a finitely generated module over formula_29 and is annihilated only by zero.) This ring is called the integral closure of formula_29 in formula_28.
Transitivity of integrality.
Another consequence of the above equivalence is that "integrality" is transitive, in the following sense. Let formula_40 be a ring containing formula_28 and formula_42. If formula_43 is integral over "formula_28" and "formula_28" integral over formula_29, then formula_43 is integral over formula_29. In particular, if formula_40 is itself integral over "formula_28" and "formula_28" is integral over formula_29, then formula_40 is also integral over formula_29.
Integral closed in fraction field.
If formula_29 happens to be the integral closure of formula_29 in "formula_28", then "A" is said to be integrally closed in "formula_28". If formula_28 is the total ring of fractions of formula_29, (e.g., the field of fractions when formula_29 is an integral domain), then one sometimes drops the qualification "in formula_28" and simply says "integral closure of formula_29" and "formula_29 is integrally closed." For example, the ring of integers formula_65 is integrally closed in the field formula_66.
Transitivity of integral closure with integrally closed domains.
Let "A" be an integral domain with the field of fractions "K" and "A' " the integral closure of "A" in an algebraic field extension "L" of "K". Then the field of fractions of "A' " is "L". In particular, "A' " is an integrally closed domain.
Transitivity in algebraic number theory.
This situation is applicable in algebraic number theory when relating the ring of integers and a field extension. In particular, given a field extension formula_67 the integral closure of formula_65 in formula_69 is the ring of integers formula_70.
Remarks.
Note that transitivity of integrality above implies that if formula_28 is integral over formula_29, then formula_28 is a union (equivalently an inductive limit) of subrings that are finitely generated formula_29-modules.
If formula_29 is noetherian, transitivity of integrality can be weakened to the statement:
Relation with finiteness conditions.
Finally, the assumption that formula_29 be a subring of formula_28 can be modified a bit. If formula_81 is a ring homomorphism, then one says formula_82 is integral if formula_28 is integral over formula_84. In the same way one says formula_82 is finite (formula_28 finitely generated formula_29-module) or of finite type (formula_28 finitely generated formula_29-algebra). In this viewpoint, one has that
Or more explicitly,
Integral extensions.
Cohen-Seidenberg theorems.
An integral extension "A" ⊆ "B" has the going-up property, the lying over property, and the incomparability property (Cohen–Seidenberg theorems). Explicitly, given a chain of prime ideals formula_97 in "A" there exists a formula_98 in "B" with formula_99 (going-up and lying over) and two distinct prime ideals with inclusion relation cannot contract to the same prime ideal (incomparability). In particular, the Krull dimensions of "A" and "B" are the same. Furthermore, if "A" is an integrally closed domain, then the going-down holds (see below).
In general, the going-up implies the lying-over. Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over".
When "A", "B" are domains such that "B" is integral over "A", "A" is a field if and only if "B" is a field. As a corollary, one has: given a prime ideal formula_100 of "B", formula_100 is a maximal ideal of "B" if and only if formula_102 is a maximal ideal of "A". Another corollary: if "L"/"K" is an algebraic extension, then any subring of "L" containing "K" is a field.
Applications.
Let "B" be a ring that is integral over a subring "A" and "k" an algebraically closed field. If formula_103 is a homomorphism, then "f" extends to a homomorphism "B" → "k". This follows from the going-up.
Geometric interpretation of going-up.
Let formula_104 be an integral extension of rings. Then the induced map
is a closed map; in fact, formula_106 for any ideal "I" and formula_107 is surjective if "f" is injective. This is a geometric interpretation of the going-up.
Geometric interpretation of integral extensions.
Let "B" be a ring and "A" a subring that is a noetherian integrally closed domain (i.e., formula_108 is a normal scheme.) If "B" is integral over "A", then formula_109 is submersive; i.e., the topology of formula_108 is the quotient topology. The proof uses the notion of constructible sets. (See also: torsor (algebraic geometry).)
Integrality, base-change, universally-closed, and geometry.
If formula_28 is integral over formula_29, then formula_113 is integral over "R" for any "A"-algebra "R". In particular, formula_114 is closed; i.e., the integral extension induces a "universally closed" map. This leads to a geometric characterization of integral extension. Namely, let "B" be a ring with only finitely many minimal prime ideals (e.g., integral domain or noetherian ring). Then "B" is integral over a (subring) "A" if and only if formula_114 is closed for any "A"-algebra "R". In particular, every proper map is universally closed.
Galois actions on integral extensions of integrally closed domains.
Proof. Suppose formula_118 for any formula_119 in "G". Then, by prime avoidance, there is an element "x" in formula_120 such that formula_121 for any formula_119. "G" fixes the element formula_123 and thus "y" is purely inseparable over "K". Then some power formula_124 belongs to "K"; since "A" is integrally closed we have: formula_125 Thus, we found formula_124 is in formula_127 but not in formula_128; i.e., formula_129.
Application to algebraic number theory.
The galois group formula_130 then acts on all of the prime ideals formula_131 lying over a fixed prime ideal formula_132. That is, ifformula_133then there is a Galois action on the set formula_134. This is called the Splitting of prime ideals in Galois extensions.
Remarks.
The same idea in the proof shows that if formula_67 is a purely inseparable extension (need not be normal), then formula_109 is bijective.
Let "A", "K", etc. as before but assume "L" is only a finite field extension of "K". Then
Indeed, in both statements, by enlarging "L", we can assume "L" is a normal extension. Then (i) is immediate. As for (ii), by the going-up, we can find a chain formula_140 that contracts to formula_141. By transitivity, there is formula_142 such that formula_143 and then formula_144 are the desired chain.
Integral closure.
Let "A" ⊂ "B" be rings and "A' " the integral closure of "A" in "B". (See above for the definition.)
Integral closures behave nicely under various constructions. Specifically, for a multiplicatively closed subset "S" of "A", the localization "S"−1"A' " is the integral closure of "S"−1"A" in "S"−1"B", and formula_145 is the integral closure of formula_146 in formula_147. If formula_148 are subrings of rings formula_149, then the integral closure of formula_150 in formula_151 is formula_152 where formula_153 are the integral closures of formula_148 in formula_155.
The integral closure of a local ring "A" in, say, "B", need not be local. (If this is the case, the ring is called unibranch.) This is the case for example when "A" is Henselian and "B" is a field extension of the field of fractions of "A".
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".
Let "A" be an formula_156-graded subring of an formula_156-graded ring "B". Then the integral closure of "A" in "B" is an formula_156-graded subring of "B".
There is also a concept of the integral closure of an ideal. The integral closure of an ideal formula_159, usually denoted by formula_160, is the set of all elements formula_161 such that there exists a monic polynomial 
with formula_163 with formula_164 as a root. Note this is the definition that appears, for example, in Eisenbud and is different from Bourbaki's and Atiyah–MacDonald's definition.
For noetherian rings, there are alternate definitions as well.
The notion of integral closure of an ideal is used in some proofs of the going-down theorem.
Conductor.
Let "B" be a ring and "A" a subring of "B" such that "B" is integral over "A". Then the annihilator of the "A"-module "B"/"A" is called the "conductor" of "A" in "B". Because the notion has origin in algebraic number theory, the conductor is denoted by formula_170. Explicitly, formula_171 consists of elements "a" in "A" such that formula_172. (cf. idealizer in abstract algebra.) It is the largest ideal of "A" that is also an ideal of "B". 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 formula_175 (i.e., "A" is the coordinate ring of the affine curve formula_176.) "B" is the integral closure of "A" in formula_177. The conductor of "A" in "B" is the ideal formula_178. More generally, the conductor of formula_179, "a", "b" relatively prime, is formula_180 with formula_181.
Suppose "B" is the integral closure of an integral domain "A" in the field of fractions of "A" such that the "A"-module formula_182 is finitely generated. Then the conductor formula_171 of "A" is an ideal defining the support of formula_182; thus, "A" coincides with "B" in the complement of formula_185 in formula_186. In particular, the set formula_187, the complement of formula_185, is an open set.
Finiteness of integral closure.
An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results.
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. This is a consequence of the Krull–Akizuki theorem. In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian. A nicer statement is this: the integral closure of a noetherian domain is a Krull domain (Mori–Nagata theorem). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain.
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 formula_189 of "A" in "L" is a finitely generated "A"-module. 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 formula_189 of "A" in "L" is a finitely generated "A"-module and is also a finitely generated "k"-algebra. The result is due to Noether and can be shown using the Noether normalization lemma as follows. 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; thus, assume "L"/"K" is purely inseparable. By the normalization lemma, "A" is integral over the polynomial ring formula_191. Since "L"/"K" is a finite purely inseparable extension, there is a power "q" of a prime number such that every element of "L" is a "q"-th root of an element in "K". Let formula_192 be a finite extension of "k" containing all "q"-th roots of coefficients of finitely many rational functions that generate "L". Then we have: formula_193 The ring on the right is the field of fractions of formula_194, which is the integral closure of "S"; thus, contains formula_189. Hence, formula_189 is finite over "S"; a fortiori, over "A". The result remains true if we replace "k" by Z.
The integral closure of a complete local noetherian domain "A" in a finite extension of the field of fractions of "A" is finite over "A". More precisely, for a local noetherian ring "R", we have the following chains of implications:
Noether's normalization lemma.
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 "y"1, "y"2, ..., "y""m" in "A" that are algebraically independent over "K" such that "A" is finite (and hence integral) over "B" = "K"["y"1..., "y""m"]. Thus the extension "K" ⊂ "A" can be written as a composite "K" ⊂ "B" ⊂ "A" where "K" ⊂ "B" is a purely transcendental extension and "B" ⊂ "A" is finite.
Integral morphisms.
In algebraic geometry, a morphism formula_203 of schemes is "integral" if it is affine and if for some (equivalently, every) affine open cover formula_204 of "Y", every map formula_205 is of the form formula_206 where "A" is an integral "B"-algebra. 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.
Absolute integral closure.
Let "A" be an integral domain and "L" (some) algebraic closure of the field of fractions of "A". Then the integral closure formula_207 of "A" in "L" is called the absolute integral closure of "A". It is unique up to a non-canonical isomorphism. The ring of all algebraic integers is an example (and thus formula_207 is typically not noetherian.)