Algebraic number

An algebraic number is a complex number that is a root of a nonzero polynomial with integer coefficients. Numbers that are not algebraic are called transcendental.

Polynomial: An expression built from variables and coefficients using addition, subtraction, multiplication, and nonnegative integer powers. Integer-coefficient polynomials provide the equations whose roots define algebraic numbers.
Degree of an algebraic number: The degree of the minimal polynomial of an algebraic number over the rational numbers. Degree measures the complexity of the number's simplest rational-coefficient equation.
Transcendental number: A complex number that is not a root of any nonzero polynomial with integer coefficients. It is precisely the alternative to being algebraic.
Abelian extension: A Galois field extension whose Galois group is abelian. Algebraic numbers in abelian extensions are central to explicit class field theory.
Ancient Greek mathematics: The mathematical traditions developed in the Greek-speaking world from antiquity through late antiquity. Greek work on incommensurable magnitudes foreshadowed distinctions among kinds of numbers.
Root of a polynomial: A value that makes a polynomial equal zero when substituted for its variable. Being a root of an integer-coefficient polynomial is the defining condition for algebraicity.
Algebraic closure: An algebraically closed field extension in which every polynomial over the original field has a root. The algebraic numbers form the algebraic closure of the rational numbers inside the complex numbers.
Rational number: A number expressible as the quotient of two integers with nonzero denominator. Every rational number is algebraic, satisfying a linear integer-coefficient equation.
Straightedge and compass construction: A geometric construction using only an unmarked straightedge and compass. Constructible coordinates are algebraic numbers obtained through successive quadratic extensions.
Évariste Galois: A French mathematician whose work founded the group-theoretic study of polynomial equations. His theory explains when algebraic equations can be solved by radicals.

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