Algebraic integer

An algebraic integer is a complex number α that satisfies a monic polynomial equation xⁿ + a_n−1xⁿ⁻¹ + … + a₀ = 0 with integer coefficients a_i.¹ Equivalently, α is an algebraic integer if its minimal polynomial over ℚ has integer coefficients and is monic.² The set of all algebraic integers forms a commutative ring under ordinary addition and multiplication, denoted ℤ̄. This ring is integrally closed in the field of all algebraic numbers, meaning that any root of a monic polynomial with algebraic-integer coefficients is itself an algebraic integer.³ A key property is that the only rational algebraic integers are the ordinary integers ℤ.⁴ For example, √2 is an algebraic integer (root of x²−2), while 1/2 is not (its minimal polynomial is 2x−1, which is not monic).

In a number field K (a finite extension of ℚ), the algebraic integers contained in K form a ring called the ring of integers, often denoted O_K.⁵ This ring is a Dedekind domain, which guarantees unique factorization of ideals into prime ideals, even when the ring itself is not a unique factorization domain.⁶ For instance, in ℚ(√−5), the ring of integers is ℤ[√−5], where 6 factors as 2·3 and (1+√−5)(1−√−5), showing non-unique factorization of elements.⁷ The study of these rings leads to the ideal class group, which measures the failure of unique factorization, and to the notion of the discriminant and the regulator. Algebraic integers are also essential in the proof of Fermat's Last Theorem, where the cyclotomic integers (roots of unity) play a central role in Kummer's work.

Beyond the classical theory, algebraic integers have surprising connections. For example, the set of algebraic integers is countable, yet it is dense in the complex plane, meaning that every complex number can be approximated arbitrarily closely by algebraic integers. A lesser-known result is that the ring of all algebraic integers is a Bézout domain, but not a principal ideal domain, and it is not Noetherian. Another edge case: an algebraic integer can be a root of a non-monic polynomial with integer coefficients, but its minimal polynomial is always monic with integer coefficients. The concept also extends to algebraic integers in function fields, where the analogy with number fields yields deep results in arithmetic geometry. Historically, the term was introduced by Dedekind in the 19th century, building on the work of Gauss and Eisenstein on cyclotomic integers.

In computational number theory, algorithms to compute rings of integers and their properties are implemented in systems like PARI/GP and SageMath. These algorithms rely on the LLL lattice reduction to find minimal polynomials and to compute integral bases. Algebraic integers appear in cryptography, particularly in pairing-based cryptography where the ring of integers of cyclotomic fields is used to construct efficient pairings. They also arise in coding theory, where algebraic-integer codes are used to design space-time codes for wireless communication. The concept of algebraic integers is also used in the theory of modular forms and in the proof of the Mordell conjecture by Faltings, which relies on properties of rings of integers in number fields.

Monic polynomial

A polynomial whose leading coefficient is 1.

Minimal polynomial

The monic polynomial of least degree with rational coefficients that has the algebraic number as a root.

Ring of integers

The set of all algebraic integers contained in a number field.

Dedekind domain

An integral domain in which every nonzero proper ideal factors uniquely into prime ideals.

Ideal class group

The group of fractional ideals modulo principal ideals, measuring the failure of unique factorization.