splitting field
短语词形变化
释义与例句
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1.
(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.
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2.
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.
数学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.
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. Saltman, Lance W. Small (editors), Selected Papers of S. A. Amitsur with Commentary, Part 2, American Mathematical Society, page 199, 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 study of the set of all splitting fields of a given c.s.a. 𝔄.
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3.
(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.
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4.
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).
数学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 of the polynomial f(x).
2001, T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117, 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.