Algebraic NumbersEdit
Algebraic numbers occupy a central place in number theory and its connections to algebra, geometry, and computation. Broadly speaking, they are the complex numbers that satisfy a polynomial equation with integer coefficients. This simple definition belies a rich structure: a countable, well-behaved field sitting inside the complex plane, containing the rational numbers and closed under the usual arithmetic operations. The standard notation for this set is the algebraic closure of the rationals, often denoted by algebraic closure in literature, and it is the locus where many classical questions about solvability of equations, field extensions, and arithmetic reflect their most concrete form.
From a practical and historical standpoint, algebraic numbers are the bridge between the concrete world of polynomials and the more abstract realm of field theory. They emerge naturally when solving polynomial equations by radicals, when studying Diophantine equations, and when examining how numbers can extend the rational field. Their behavior is tightly controlled by the minimal polynomial of each element: a monic irreducible polynomial with rational coefficients, whose roots—called the conjugates—sit in a finite set determined by the degree of the extension. This interplay underwrites a great deal of the modern theory of numbers, including how one measures size, how one tracks prime decomposition in extensions, and how one understands symmetries through the lens of Galois theory.
Definition and basic properties
An algebraic number is a complex number a for which there exists a nonzero Polynomial P with integer coefficients such that P(a) = 0. Equivalently, a is algebraic over the field of rational numbers Rational number.
The collection of all algebraic numbers is denoted by algebraic number and forms a field. In particular, if a and b are algebraic, so are a + b, a − b, a·b, and a/b whenever b ≠ 0. Thus, algebraic numbers are closed under the basic arithmetic of a field, and every algebraic number is contained in some finite field (algebra) extension of Q.
The degree of an algebraic number a over Q is the degree of its minimal polynomial, and this degree equals the dimension of the vector space Q(a) over Q. All conjugates of a are the other roots of that minimal polynomial.
The set of algebraic numbers is countable, yet dense in the real line and in the complex plane because the rational numbers themselves are dense and rational numbers are a subset of algebraic numbers.
The algebraic numbers form the smallest field containing Q that is closed under taking roots of polynomials with coefficients in that field; in this sense, they are the algebraic closure of Q within the complex numbers.
Algebraic numbers and algebraic integers
An algebraic integer is an algebraic number that is a root of a monic Polynomial with integer coefficients. Not every algebraic number is an algebraic integer, but the algebraic integers form a ring, denoted Algebraic integer in a given number field K.
A key object is a number field, a finite extension of Q. Every algebraic number lies in some number field, and the ring of integers of that field, together with ideals and units, encodes essential arithmetic features of the field. The study of these rings—how primes factor, how units behave, and how the discriminant governs ramification—has been a cornerstone of modern arithmetic.
Important tools include norms and traces, discriminants, and ramification theory. The ring of integers O_K is the integral closure of Z in K and serves as the natural setting for questions about factorization and arithmetic in K. Dirichlet’s unit theorem describes the structure of the unit group of O_K, and the discriminant measures how the arithmetic of the field sits inside its embedding into the complex numbers.
For a concrete view, one may consider simple examples like Number field and its ring of integers, or the more complicated geometry of curves and surfaces arising from higher-degree extensions. These ideas connect with classical problems such as solutions to Diophantine equations and the distribution of primes in extensions.
Transcendence versus algebraicity
The real line and the complex plane contain far more numbers than the algebraic ones. In fact, almost all complex numbers are transcendental, meaning they do not satisfy any nonzero polynomial with rational coefficients. This highlights the special status of the algebraic numbers: they are a discrete, countable universe within the continuum.
Some numbers are famous examples of algebraicity: for instance, sqrt(2) is a root of x^2 − 2, and many other familiar square roots and roots of low-degree polynomials lie in this class. On the other hand, numbers such as pi and e have been proven to be transcendental, a result with deep consequences for geometry and analysis. The boundary between algebraic and transcendental numbers marks a fundamental line in the landscape of real and complex numbers.
The study of transcendence often interacts with algebraic number theory through questions about values of special functions at algebraic points, or through results like the Lindemann–Weierstrass theorem, which connects the nature of certain exponentials to the transcendence of numbers arising from algebraic inputs.
Computation, algorithms, and applications
Computation with algebraic numbers relies on algorithms for manipulating polynomials, minimal polynomials, and field embeddings. Techniques from computational algebra, including lattice-based methods, enable explicit calculations with algebraic numbers and with the rings of integers in number fields.
In practice, this theory underpins modern computational number theory, which in turn supports applications in cryptography, computer algebra systems, and digital security. Algorithms for primality testing, integer factorization, and operations in finite and infinite extensions of Q are all linked to the arithmetic of algebraic numbers and their integer analogues.
The arithmetic of algebraic numbers also informs the study of Diophantine equations. Classic problems such as Pell’s equation live in the environment of quadratic fields, where the units of the ring of integers can be described explicitly, and where continued fractions provide practical algorithms for solutions.
Philosophical and methodological notes
A long-running debate in mathematics concerns whether mathematical truth is discovered or invented. From a tradition-minded perspective, the theory of algebraic numbers embodies a reliable, highly structured form of reasoning about solvability of equations and the arithmetic of field extensions. It reflects a disciplined tradition of deduction, classification, and explicit construction.
Related discussions touch on foundations: the role of the axiom of choice, the balance between constructive versus non-constructive proofs, and the search for proofs that yield explicit algorithms. In algebraic number theory, many results rely on tools that are non-constructive in nature, while there is also a strong thread of constructive methods in explicit class field theory and algorithmic number theory.
Critics from various quarters argue about the direction and priorities of mathematical research, including the place of pure theory versus applied outcome and the need to diversify the mathematical conversation. Proponents of traditional, rigorous approaches emphasize that deep structural results in algebraic numbers have proven essential for technology and science, even if some modern developments rely on highly abstract machinery.