splitting field

Noun

• 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); given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.
• 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).
• 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.
• A field K over which a K-representation of G exists which includes the character χ; a field over which a K-representation of G exists which includes every irreducible character in G.