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| Int(g) | |
|---|---|
| Name | Int(g) |
| Definition | Integral closure of a subring generated by an element g |
| Type | Ring-theoretic construction |
| Field | Algebra |
| Related | Integral closure, Minimal polynomial, Discriminant, Norm |
Int(g)
Int(g) denotes the integral closure or integral span associated to an algebraic element g in a ring or an extension; it appears in algebraic number theory, commutative algebra, and algebraic geometry. In typical usage Int(g) designates the subring consisting of elements integral over a base ring generated by g, often coinciding with the ring of integers in a finite extension generated by a single algebraic element. Instances of Int(g) arise in the study of number fields such as Q(√d), coordinate rings of affine varieties like A^1, and module structures in the representation theory of GL_n.
For a commutative domain R and an element g in an R-algebra S, Int(g) is defined as the subring of S consisting of elements that are integral over the subring R[g] generated by g. Concretely, an element x belongs to Int(g) if there exists a monic polynomial f(X) in R[g][X] with f(x)=0. When R is the ring of integers Z and g is algebraic over Q, Int(g) often equals the integral closure of Z[g] in the field extension Q(g). Typical contexts include algebraic number fields such as Q(√2), cyclotomic fields like Q(ζ_n), and function fields of curves such as those of P^1 over finite fields.
Int(g) is integrally closed in the subfield or subring it generates: it is the full integral closure of R[g] inside a chosen ambient integral domain. It is a finitely generated R-module when the extension R[g]→S is finite; this connects to the Noether normalization lemma, the primitive element theorem for separable extensions, and the computation of the discriminant and norm. The ring Int(g) is stable under addition and multiplication and contains R[g]; it coincides with R[g] exactly when the minimal polynomial of g over the field of fractions of R has coefficients in R and the index of R[g] in the full ring of integers equals one, a condition detected by divisibility of the discriminant relative to primes of Spec R. Localizing at a prime ideal p of R gives (Int(g))_p equal to the integral closure of R_p[g] in the local extension, relating to Dedekind domain factorization and ramification at p. In Galois extensions generated by g where the Galois group is isomorphic to groups such as S_n or C_n, Int(g) inherits Galois module structure and can be studied via the action of inertia and decomposition groups.
Classical examples include rings of integers of quadratic fields: for g = √d one compares Int(√d) with Z[√d] to decide whether the ring of integers of Q(√d) equals Z[√d] or requires the half-integer (1+√d)/2 as in discriminant congruences mod 4 related to Gauss. For cubic fields generated by a root g of an irreducible cubic f(X) in Z[X], computing Int(g) uses the index formula involving the discriminant of f and may involve integral bases found via algorithms developed by Dedekind and refined by Lenstra and Belabas. In cyclotomic examples with g = ζ_n, Int(ζ_n) is the full ring of integers Z[ζ_n] of the cyclotomic field Q(ζ_n), a fact tied to Kronecker–Weber theorem and to explicit Gauss sums. For algebraic function fields, take g equal to a coordinate on a plane curve defined over F_q: Int(g) coincides with the coordinate ring of the normalization of the curve in its function field, a computation connected to resolution of singularities studied by Zariski and Abhyankar.
Int(g) is crucial in algebraic number theory for determining rings of integers of number fields generated by single elements, which impacts computation of class groups of Dedekind domains, unit groups via Dirichlet's unit theorem, and explicit factorization of ideals used in algorithms by Lenstra, Cohen, and Buchmann. In algebraic geometry Int(g) models normalization of affine varieties and is used in desingularization techniques linked to Hironaka and to the study of curve singularities by Puiseux expansions. In arithmetic dynamics and Diophantine equations, rings of integral elements generated by dynamical quantities relate to Mordell–Weil theorem applications and height bounds developed by Faltings and Baker. Computational algebra systems employ Int(g) algorithms for computing integral closures, as implemented in packages influenced by the work of van Hoeij and PARI/GP.
Variants include Int_S(g), the integral closure of R[g] inside a specified overring S, and relative integral closures for extensions of Dedekind domains such as O_K inside O_L for number fields K⊂L. One generalizes from single-generator cases to multi-generator integral closures for finitely generated algebras R[g_1,...,g_m], linking to normalization algorithms of affine algebras and to the study of coordinate rings of normalization morphisms used by Serre and Grothendieck in scheme theory. Noncommutative analogues consider integral dependence in orders of central simple algebras studied in the context of Brauer group elements and maximal orders in division algebras.
The study of rings generated by algebraic elements and their integral closures traces to 19th-century work on algebraic integers by Kummer, Dedekind, and Kummer's contemporaries in pursuit of Fermat's Last Theorem. Dedekind formalized rings of integers and introduced discriminants and factorization of primes, while later contributors like Hecke, Leopoldt, and Kronecker developed explicit descriptions for cyclotomic and quadratic cases. Algorithmic determination of Int(g) advanced in the 20th century through contributions of Hasse, Noether, and computational pioneers such as Lenstra and Cohen, culminating in practical algorithms and software in the late 20th and early 21st centuries used by researchers in algebraic number theory and computational algebraic geometry.