splitting field

Theorem 3.2. If K is a field and f∈K[x] has degree n>1, then there exists a splitting field F of f with [F:K]<n!

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↦L such that A⊗_KL is split; in the special case A=K[x]/f(x) this is the same as a splitting field of the polynomial f(x).

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.

DEFINITION 2. A field K is called a splitting field of a character 𝜒 of a group G if 𝜒∈ operatorname Char_K(G), i.e., 𝜒 is afforded by a K-representation of G.

Let T be a representation of G affording the character 𝜒. It follows from Definition 2 that K is a splitting field of 𝜒 if and only if T is equivalent to 𝛥, where 𝛥 is a K-representation of G. In other words, K is a splitting field of a character 𝜒 if and only if a representation T affording 𝜒 is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters 𝜒₁,𝜒₂, then K is a splitting field of 𝜒₁+𝜒₂, Therefore, in studying splitting fields, we may consider irreducible characters only.

DEFINITION 3. A field K is called a splitting field of a group G if it is a splitting field for every 𝜒∈ operatorname Irr(G).