splitting field

Noun

 * 1)   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);  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)  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).
 * 3) * 2001,, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117,
 * Ex. 7.6. For a finite-dimensional $$k$$-algebra $$R$$, let $$T(R)=\operatorname{rad} R +[R,R]$$, where $$[R,R]$$ denotes the subgroup of $$R$$ generated by $$ab-ba$$ for all $$a,b\in R$$. Assume that $$k$$ has characteristic $$p>0$$. Show that
 * $$T(R)\subseteq\{a\in R: a^{p^m}\text{ for some }m\ge 1\}$$,
 * with equality if $$k$$ is a splitting field for $$R$$.
 * 1)  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.
 * 2) * 1955,, Generic Splitting Fields of Central Simple Algebras, , 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, , 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 $$\mathfrak C$$ are the extensions of $$\mathfrak C$$ that split the algebras. A field $$\mathfrak F\supseteq\mathfrak C$$ is said to split a c.s.a. $$\mathfrak A$$ if $$\mathfrak A\otimes\mathfrak F$$ is a total matrix ring over $$\mathfrak F$$. The present study is devoted to the study of the set of all splitting fields of a given c.s.a. $$\mathfrak A$$.
 * 1)   A 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.
 * 2) * 1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2,, page 165,
 * D EFINITION  2. A field $$K$$ is called a splitting field of a character $$\chi$$ of a group $$G$$ if $$\chi\in\operatorname{Char}_K(G)$$, i.e., $$\chi$$ is afforded by a $$K$$-representation of $$G$$.
 * Let $$T$$ be a representation of $$G$$ affording the character $$\chi$$. It follows from Definition 2 that $$K$$ is a splitting field of $$\chi$$ if and only if $$T$$ is equivalent to $$\Delta$$, where $$\Delta$$ is a $$K$$-representation of $$G$$. In other words, $$K$$ is a splitting field of a character $$\chi$$ if and only if a representation $$T$$ affording $$\chi$$ 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 $$\chi_1,\chi_2,$$ then $$K$$ is a splitting field of $$\chi_1+\chi_2$$, Therefore, in studying splitting fields, we may consider irreducible characters only.
 * D EFINITION  3. A field $$K$$ is called a splitting field of a group $$G$$ if it is a splitting field for every $$\chi\in\operatorname{Irr}(G)$$.
 * 1) * 1955,, Generic Splitting Fields of Central Simple Algebras, , 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, , 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 $$\mathfrak C$$ are the extensions of $$\mathfrak C$$ that split the algebras. A field $$\mathfrak F\supseteq\mathfrak C$$ is said to split a c.s.a. $$\mathfrak A$$ if $$\mathfrak A\otimes\mathfrak F$$ is a total matrix ring over $$\mathfrak F$$. The present study is devoted to the study of the set of all splitting fields of a given c.s.a. $$\mathfrak A$$.
 * 1)   A 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.
 * 2) * 1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2,, page 165,
 * D EFINITION  2. A field $$K$$ is called a splitting field of a character $$\chi$$ of a group $$G$$ if $$\chi\in\operatorname{Char}_K(G)$$, i.e., $$\chi$$ is afforded by a $$K$$-representation of $$G$$.
 * Let $$T$$ be a representation of $$G$$ affording the character $$\chi$$. It follows from Definition 2 that $$K$$ is a splitting field of $$\chi$$ if and only if $$T$$ is equivalent to $$\Delta$$, where $$\Delta$$ is a $$K$$-representation of $$G$$. In other words, $$K$$ is a splitting field of a character $$\chi$$ if and only if a representation $$T$$ affording $$\chi$$ 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 $$\chi_1,\chi_2,$$ then $$K$$ is a splitting field of $$\chi_1+\chi_2$$, Therefore, in studying splitting fields, we may consider irreducible characters only.
 * D EFINITION  3. A field $$K$$ is called a splitting field of a group $$G$$ if it is a splitting field for every $$\chi\in\operatorname{Irr}(G)$$.

Usage notes

 * The polynomial (respectively, central simple algebra or character) is said to over its splitting field.
 * More formally, the smallest extension field $$L$$ of $$K$$ such that $$\textstyle p(X) = c\prod_{i=1}^{\deg(p)}(X - a_i)$$ where $$c\in K$$ and, for each $$i$$, $$(X - a_i) \in L[X]$$.
 * Perhaps more simply, $$L$$ is the smallest extension of $$K$$ in which every root of $$p$$ is an element. (Note that the selected definition, in contrast, refers explicitly to the factorisation of the polynomial.)
 * An extension $$L$$ that is a splitting field for some set of polynomials over $$K$$ is called a of $$K$$.
 * An extension $$L$$ that is a splitting field for some set of polynomials over $$K$$ is called a of $$K$$.

Translations

 * Basque: deskonposizio gorputz, banatze gorputz
 * French: corps de décomposition, corps des racines, corps de déploiement
 * German: Zerfällungskörper
 * Italian: campo di spezzamento, campo di riducibilità completa
 * Portuguese: corpo de decomposição, corpo de fatoração
 * Spanish: cuerpo de descomposición