Splitting field

English

Noun

1. Template:Lb Template:Lb 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); Template:Lb 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. Template:Lb 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).

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   • 2001, Template:W, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117,

     Ex. 7.6. For a finite-dimensional $k$-algebra $R$, let $T(R)=\mathrm{rad}R+[R,R]$, 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 $R$.

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3. Template:Lb 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.

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   • 1955, Template:W, Generic Splitting Fields of Central Simple Algebras, Template:W, 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, Template:W, 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 $𝔉\supseteq ℭ$ is said to split a c.s.a. $𝔄$ if $𝔄\otimes 𝔉$ 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. $𝔄$.

4. Template:Lb Template:Lb A field K over which a K-representation of G exists which includes the character χ; Template:Lb a field over which a K-representation of G exists which includes every irreducible character in G.
   • 1999, P. Shumyatsky, V. Zobina (translators), David Louvish (editor of translation), Ya. G. Berkovich, E. M. Zhmud’, Characters of Finite Groups, Volume 2, Template:W, page 165,

     DEFINITION 2. A field $K$ is called a splitting field of a character $\chi$ of a group $G$ if $\chi \in {\mathrm{Char}}_K(G)$, 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 $\mathrm{\Delta }$, where $\mathrm{\Delta }$ 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.

     DEFINITION 3. A field $K$ is called a splitting field of a group $G$ if it is a splitting field for every $\chi \in \mathrm{Irr}(G)$.

Usage notes

• The polynomial (respectively, central simple algebra or character) is said to Template:M over its splitting field.
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  • More formally, the smallest extension field $L$ of $K$ such that $p(X)=c{\prod }_{i=1}^{\mathrm{deg}(p)}(X-a_i)$ 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 Template:M of $K$.

Translations

• Basque: Template:T, Template:T
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See also

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Further reading

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  • Splitting field of a polynomial on Template:W
  • splitting field on Template:W
  • Splitting Field on Template:W
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  • Template:Pedia
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  • Notes on Splitting Fields on University of Minnesota math user home pages (unreviewed)
  • What is a split K-algebra? on Template:W
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  • character on Template:W