splitting fields

Noun

splitting field (plural splitting fields)

1. (Galois theory) (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.
   • Theorem 3.2. If K is a field and f#92;inK#91;x#93; has degree n#92;ge 1, then there exists a splitting field F of f with #91;F#58;K#93;#92;len#33;1974, Thomas W. Hungerford, Algebra, Springer, page 258

   Synonyms:

   root field

2. (ring theory) 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).
   • 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#92;mapstoL such that A#92;otimes#95;KL is split; in the special case A#61;K#91;x#93;#47;f(x) this is the same as a splitting field...
3. (ring theory) 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.
   • 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. (character theory) (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.
   • 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...
   • 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).1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2, American Mathematical Society, page 165