Splitting Field Definition

noun

(algebra SPLITTING FIELD 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.

Wiktionary

(algebra SPLITTING FIELD (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 is a field extension K\mapsto L such that A\otimes_K L is split; in the special case A=Kx/f(x) this is the same as a splitting field of the polynomial f(x).

Wiktionary

(algebra SPLITTING FIELD 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.

Wiktionary

(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.

Wiktionary