splitting field

Noun

Meaning

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.
Synonym: root field

(algebra, ring theory, of a K-algebra) Given a finite-dimensional K-algebra (algebra over a field), an extension field whose every simple (indecomposable) module is absolutely simple (remains simple after the scalar field has been extended to said extension field).
The terminology "splitting field of a K-algebra" is motivated by the same terminology regarding a polynomial. A splitting field of a K-algebra



A


{\displaystyle A}

is a field extension



K
↦
L


{\displaystyle K\mapsto L}

such that



A

⊗

K


L


{\displaystyle A\otimes _{K}L}

is split; in the special case



A
=
K
[
x
]

/

f
(
x
)


{\displaystyle A=K[x]/f(x)}

this is the same as a splitting field of the polynomial



f
(
x
)


{\displaystyle f(x)}

.

(algebra, ring theory, of a central simple algebra) Given a central simple algebra A over a field K, another field, E, such that the tensor product A⊗E is isomorphic to a matrix ring over E.
Every finite dimensional central simple algebra has a splitting field: moreover, if said CSA is a division algebra, then a maximal subfield of it is a splitting field.

(algebra, character theory) (of a character χ of a representation of a group G) A field K over which a K-representation of G exists which includes the character χ; (of a group G) a field over which a K-representation of G exists which includes every irreducible character in G.

Examples

Definition In terms of the roots, the discriminant is given by : where is the leading coefficient and are the roots (counting multiplicity ) of the polynomial in some splitting field. Source: Internet

Lang 2002, p. 292 (Theorem VI.7.2) The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Source: Internet

Close letter words and terms