Algebraic number

An algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients. Equivalently, it is a root of a non-zero polynomial with integer coefficients, since any polynomial with rational coefficients can be converted into one with integer coefficients by clearing denominators. Examples of algebraic numbers include every rational number, square roots such as √2, complex numbers such as i, and roots of unity. Complex numbers that are not algebraic are called transcendental numbers; the best-known examples are π and e.

The set of all algebraic numbers, usually denoted 𝔸 or Q̄ (as it constitutes the algebraic closure of the rational numbers ℚ inside the complex numbers ℂ), is a countable, algebraically closed field. Algebraic numbers are the central objects of algebraic number theory, and their study connects polynomial algebra, field theory, Diophantine approximation, and arithmetic geometry.

Examples

Every rational number a/b (with b ≠ 0) is algebraic, since it is a root of the linear polynomial bx − a. Further standard examples include:

• Irrational square and higher roots, such as √2 (a root of x² − 2) and ∛2 (a root of x³ − 2);
• Quadratic irrationals such as the golden ratio (1 + √5)/2, a root of x² − x − 1;
• The imaginary unit i, a root of x² + 1;
• Roots of unity, that is, complex solutions of xⁿ − 1 = 0, which include numbers such as (−1 + i√3)/2;
• Any number expressible from rational numbers using finitely many additions, subtractions, multiplications, divisions, and root extractions, such as (1 + √2 + ∛3)/7.

By contrast, the numbers e, π, the Gelfond–Schneider constant 2^(√2), and Liouville's constant Σ 10^(−k!) are known to be transcendental. For several famous constants the question remains open: it is not known whether e + π, e·π, π^e, or the Euler–Mascheroni constant γ are irrational, algebraic, or transcendental.

Historical development

The first algebraic irrational known to mathematics was √2, whose irrationality was established by the Pythagorean school in the fifth century BCE, a discovery traditionally linked to the incommensurability of the diagonal and side of a square. The systematic study of numbers arising as roots of equations, however, developed much later. In the sixteenth century, the solution of cubic and quartic equations by Italian algebraists, notably Cardano's Ars Magna (1545), forced mathematicians to work with expressions involving square roots of negative numbers, later formalized as complex numbers—all of them algebraic when built from rational inputs.

The terminology itself evolved in the eighteenth century: Leibniz had distinguished "algebraic" from "transcendental" curves, and Euler applied a similar distinction to quantities and functions. A rigorous theory of transcendental numbers began with Joseph Liouville, who in 1844 proved that transcendental numbers exist by exhibiting explicit examples and by showing that algebraic numbers cannot be approximated by rationals too rapidly.

In 1874 Georg Cantor proved that the set of algebraic numbers is countable, while the real numbers are not, showing that "almost all" real numbers are transcendental. Charles Hermite proved the transcendence of e in 1873, and Ferdinand von Lindemann extended the method to π in 1882, thereby settling the ancient problem of squaring the circle. Kurt Gelfond and Theodor Schneider independently proved the Gelfond–Schneider theorem in 1934, resolving Hilbert's seventh problem.

In a parallel development, the nineteenth century saw the birth of algebraic number theory: Ernst Kummer introduced his "ideal numbers" to restore unique factorization in cyclotomic fields, Richard Dedekind recast these as the theory of ideals in rings of integers (1871), and Leopold Kronecker, David Hilbert (whose Zahlbericht of 1897 systematized the field), and others built the modern framework of number fields.

Basic properties

The defining property can be rephrased in the language of field theory: a complex number α is algebraic if and only if the field extension ℚ(α) has finite degree over ℚ. The fundamental properties of algebraic numbers follow largely from this observation:

• Field structure. If α and β are algebraic, then so are α + β, α − β, αβ, and, when β ≠ 0, α/β. Indeed, ℚ(α, β) is a finite-dimensional vector space over ℚ, and every element of it is algebraic. Hence the algebraic numbers form a field.
• Closure under polynomial operations. If α is algebraic, then every root of any polynomial with algebraic coefficients that includes α in its construction is again algebraic; in particular, n-th roots of algebraic numbers are algebraic.
• Minimal polynomial. Every algebraic number α is the root of a unique monic polynomial irreducible over ℚ, called its minimal polynomial. The degree of this polynomial is the degree of α. For example, the degree of √2 is 2 and that of ∛2 is 3, while rational numbers have degree 1.
• Conjugates. The roots of the minimal polynomial of α are called its algebraic conjugates. For instance, the conjugates of the golden ratio are (1 + √5)/2 and (1 − √5)/2. The product of the conjugates, scaled appropriately, is the norm; for an algebraic integer, the norm is an ordinary integer.

A central theorem states that the field of algebraic numbers is algebraically closed: any polynomial whose coefficients are algebraic numbers has all of its roots in 𝔸. Thus 𝔸 is an algebraic closure of ℚ, and it is unique up to isomorphism.

The field of algebraic numbers

The field 𝔸 = Q̄ has a rich structure. It is countable, because there are only countably many polynomials with rational coefficients and each has finitely many roots. Despite being countable, 𝔸 is dense in ℂ in the usual topology, so algebraic numbers occur arbitrarily close to any complex number. Under the metric inherited from ℂ, however, 𝔸 is not complete: sequences of algebraic numbers can converge to transcendental limits, such as the partial sums of the series for e.

The group of field automorphisms of 𝔸 fixing ℚ pointwise is the absolute Galois group Gal(Q̄/ℚ), an infinite profinite group whose structure encodes deep arithmetic information and whose finite quotients correspond, via the fundamental theorem of Galois theory, to the finite extensions of ℚ.

Algebraic integers

An algebraic integer is an algebraic number that is a root of a monic polynomial (leading coefficient 1) with integer coefficients. Thus √2 and i are algebraic integers, while 1/2 and 3/2 are not, though they are algebraic. The sum and product of algebraic integers are again algebraic integers, so they form a ring, denoted 𝒪.

For each number field K (a finite extension of ℚ), the algebraic integers contained in K form a ring 𝒪_K called the ring of integers of K. Examples include ℤ itself (for K = ℚ), the Gaussian integers ℤ[i], and the Eisenstein integers ℤ[ω], where ω is a primitive cube root of unity. Rings of integers generalize the ordinary integers and inherit many of their properties, but with a crucial difference: unique factorization into primes can fail. In ℤ[√−5], for example,

6 = 2 · 3 = (1 + √−5)(1 − √−5)

are two genuinely distinct factorizations into irreducibles. Dedekind showed that unique factorization is restored at the level of ideals: every ideal in 𝒪_K factors uniquely into prime ideals. The extent of the failure of unique factorization of elements is measured by the ideal class group and its size, the class number, which is a fundamental invariant in algebraic number theory. Notably, the ring of all algebraic integers is a Bézout domain: every finitely generated ideal in it is principal.

Countability and transcendental numbers

Cantor's argument that 𝔸 is countable runs as follows: polynomials with rational coefficients can be enumerated, each non-zero polynomial has at most finitely many roots, and a countable union of finite sets is countable. Since ℝ and ℂ are uncountable, transcendental numbers not only exist but vastly outnumber algebraic ones; in the sense of Lebesgue measure, almost every real number is transcendental. Nevertheless, proving transcendence for any specific given number is typically extremely difficult.

Liouville's theorem provides the classical route: an algebraic number of degree n > 1 cannot be approximated by rationals better than a bound of order 1/qⁿ. Liouville exploited this to construct his eponymous constant, whose decimal expansion contains 1s at factorial positions, making it too well approximable to be algebraic. Hermite's proof of the transcendence of e (1873) and Lindemann's proof for π (1882) used the exponential function instead; the Hermite–Lindemann theorem states that e^α is transcendental for every non-zero algebraic α, and the stronger Lindemann–Weierstrass theorem governs linear relations among exponentials of distinct algebraic numbers.

Approximation by rational numbers

The interplay between algebraicity and rational approximation grew into the theory of Diophantine approximation. Liouville's bound was sharpened by Thue (1909) and Siegel (1921), and decisively by Klaus Roth (1955), who proved that for any algebraic irrational α and any ε > 0, the inequality |α − p/q| < 1/q^(2+ε) has only finitely many rational solutions p/q. Roth's theorem is essentially best possible, since continued fractions supply infinitely many approximations with |α − p/q| < 1/q², and it earned Roth the Fields Medal in 1958. These results have powerful consequences for Diophantine equations: Thue's theorem implies, for instance, that equations of the form F(x, y) = m, for homogeneous irreducible F of degree at least three, have only finitely many integer solutions.

Applications and significance

Solvability by radicals. Galois theory associates to each algebraic number, or each polynomial, a finite group measuring the symmetries of its conjugates. Building on the work of Abel and Ruffini (who showed the general quintic cannot be solved by radicals), Galois theory characterizes precisely when the roots of a polynomial admit expressions using arithmetic operations and root extraction: exactly when the corresponding group is solvable.

Geometric constructibility. A real number can be constructed with compass and straightedge from a unit length only if it lies in a field obtained from ℚ by a tower of quadratic extensions; in particular, its degree over ℚ must be a power of two. This criterion, developed by Wantzel, settles three classical problems: doubling the cube requires ∛2 of degree 3; trisecting a general angle leads to cos 20°, of degree 3; and squaring the circle would require constructing π, which is not even algebraic. All three constructions are therefore impossible. Gauss showed that the regular 17-gon is constructible, and the Gauss–Wantzel theorem characterizes constructible regular n-gons in terms of Fermat primes.

The Gelfond–Schneider theorem. If α and β are algebraic numbers with α ≠ 0, 1 and β irrational, then α^β is transcendental. This resolves Hilbert's seventh problem and yields the transcendence of 2^(√2) and of Gelfond's constant e^π = (−1)^(−i).

Arithmetic geometry. Algebraic numbers underpin modern number theory: the theory of L-functions, elliptic curves, and the Langlands program all concern arithmetic over number fields. A celebrated result in this vein is the Kronecker–Weber theorem, which states that every finite abelian extension of ℚ lies inside a cyclotomic field generated by a root of unity.

Computational aspects

An algebraic number can be represented exactly in a computer by its minimal polynomial together with an isolating interval (or a sufficiently tight rational interval) that distinguishes it from its conjugates—a scheme going back to Dedekind. Arithmetic on such representations is carried out using resultants and algorithms for polynomial factorization over ℚ, with the LLL lattice-reduction algorithm (1982) playing a key role in practical factorization and in recovering minimal polynomials from numerical approximations. Algebraic numbers appear as exact "Root" objects in systems such as Mathematica, Maple, SageMath, Magma, and PARI/GP, which implement extensive algorithms for number fields, ideals, and class groups. Algebraic number computations also lie at the heart of the number field sieve, the fastest known general-purpose algorithm for factoring large integers, with consequences for cryptography.

The theory of heights assigns to each algebraic number a non-negative real number measuring its arithmetic complexity; Northcott's theorem, stating that only finitely many algebraic numbers have bounded degree and bounded height, makes heights a fundamental tool in Diophantine geometry.

Generalizations and open problems

The definition extends verbatim to arbitrary base fields: an element α of an extension field L is algebraic over a field K if it is a root of a non-zero polynomial with coefficients in K. The algebraic closure of a field, its existence and uniqueness up to isomorphism, is a cornerstone of field theory. Over the p-adic numbers, one obtains the algebraic closure Q̄_p and its completion ℂ_p, which is itself algebraically closed; Ostrowski's theorem shows that the usual absolute value and the p-adic absolute values are the only non-trivial absolute values on ℚ, so that algebraic numbers can be studied simultaneously in