Lineaarteguritega

Lineaarteguritega is a term used in algebra to describe polynomials that factor completely into linear factors. For a polynomial f in F[x], where F is a field, lineaarteguritega means there exists an extension field E ⊇ F such that f(x) = a_n ∏_{i=1}^n (x − α_i) with α_i ∈ E. The α_i are the roots of f in E, counted with multiplicity. If all α_i lie in F, then f splits completely over F and is sometimes simply said to split in F or be split over F.

Examples illustrate the concept. Over the real numbers, the cubic x^3 − 6x^2 + 11x − 6 factors as (x − 1)(x − 2)(x − 3), so it is lineaarteguritega over R. The polynomial x^2 + 1 has no linear factors over the rationals, but over the complex numbers it factors as (x − i)(x + i), so it is lineaarteguritega over C.

Relation to splitting fields and roots shows why the notion matters. The smallest field over which f splits completely is called its splitting field. The ability to express f as a product of linear factors is connected to the action of the Galois group permuting the roots. Distinct linear factors correspond to simple roots; repeated roots occur when the discriminant vanishes.

Applications include factorization, solving polynomial equations, and understanding the structure of roots. In computational algebra, algorithms aim to factor polynomials into linear factors over various coefficient fields. See also: polynomial, splitting field, Galois theory.