splitting field

Meanings

1. (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.
2. 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).
3. 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.
4. (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.