Algebraic integer

An algebraic integer is a complex number that satisfies a monic polynomial with integer coefficients. These numbers form a ring and generalize the ordinary integers within algebraic number theory.

Monic polynomial: A polynomial whose leading coefficient is one. The defining equation must have leading coefficient one.
Ring of integers: The set of algebraic integers in a number field, forming a ring denoted by its field's script O. It collects the algebraic integers belonging to each number field.
Unique factorization domain: An integral domain in which every nonzero nonunit factors uniquely into irreducibles, up to units and order. Rings of integers need not have unique factorization of elements.
Ernst Kummer: A nineteenth-century German mathematician who developed ideal numbers to address failures of unique factorization. His work on cyclotomic integers helped shape modern algebraic number theory.
Transcendental number: A complex number that is not a root of any nonzero polynomial with integer coefficients. It lies outside the algebraic numbers, and therefore outside the algebraic integers.
Minimal polynomial: The monic irreducible polynomial over a field that has a given algebraic element as a root. Its coefficients show whether an algebraic number is an algebraic integer.
Trace (algebra): For a finite field extension, the trace of an element is the sum of its conjugates, counted with multiplicity. An algebraic integer has an integer trace in a number field.
Ideal class group: A finite group measuring the failure of unique factorization of ideals in a ring of integers. It captures factorization failures that arise in algebraic-integer rings.
Richard Dedekind: A German mathematician who founded major parts of modern algebra and number theory. He formalized rings of integers and ideals in number fields.
Rational integer: An element of the familiar set of integers, usually denoted ℤ. Every rational integer is an algebraic integer, giving the basic examples.

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