splitting field

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
   • In the case of the polynomial t²#43;1#92;in#92;R#91;t#93;, the situation is quite clear; its splitting field is #92;C since t²#43;1#61;(t#43;i)(t-i) where i#61;#92;sqrt#123;-1#125;.2003, Derek J. S. Robinson, An Introduction to Abstract Algebra, Walter de Gruyter, page 130

   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...
   • Ex. 7.6. For a finite-dimensional k-algebra R, let T(R)= operatorname radR+[R,R], where [R,R] denotes the subgroup of R generated by ab-ba for all a,b∈R. Assume that k has characteristic p>0. Show that
     T(R)⊆a∈R:aforsomem>1,
     with equality if k is a splitting field for R.2001, T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117
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.
   • The main tool in studying the structure of division algebras, or more generally, of central simple algebras (c.s.as) over a field ℭ are the extensions of ℭ that split the algebras. A field 𝔉⊇ℭ is said to split a c.s.a. 𝔄 if 𝔄⊗𝔉 is a total matrix ring over 𝔉. The present study is devoted to the...1955, Shimshon A. Amitsur, Generic Splitting Fields of Central Simple Algebras, Annals of Mathematics, Volume 62, Number 1, Reprinted in 2001, Avinoam Mann, Amitai Regev, Louis Rowen, David J....
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