Algebraic extension in the context of "Algebraic number field"

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

In mathematics, an algebraic number field (or simply number field) is an extension field ${\displaystyle K}$ of the field of rational numbers ${\displaystyle \mathbb {Q} }$ such that the field extension ${\displaystyle K/\mathbb {Q} }$ has finite degree (and hence is an algebraic field extension).Thus ${\displaystyle K}$ is a field that contains ${\displaystyle \mathbb {Q} }$ and has finite dimension when considered as a vector space over ${\displaystyle \mathbb {Q} }$.

The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.

In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems.

The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: