Definitions
1.NounGiven 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.
2.NounGiven 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)}
."
3.NounGiven 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."
4.NounA 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.