splitting field

English

Noun

splitting field (plural splitting fields)

1. (algebra, 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.

   Synonym: root field

   • 1974, Thomas W. Hungerford, Algebra, Springer, page 258:

     Theorem 3.2. If $K$ is a field and $f\in K[x]$ has degree $n\geq 1$, then there exists a splitting field $F$ of $f$ with $[F:K]\leq n!$

   • 2003, Derek J. S. Robinson, An Introduction to Abstract Algebra, Walter de Gruyter, page 130:

     In the case of the polynomial $t^{2}+1\in \mathbb {R} [t]$, the situation is quite clear; its splitting field is $\mathbb {C}$ since $t^{2}+1=(t+i)(t-i)$ where $i={\sqrt {-1}}$.

   • 2009, Paolo Aluffi, Algebra: Chapter 0, American Mathematical Society, page 430:

     Example 4.3. By definition, $\mathbb {Q} (i)$ is the splitting field of $x^{2}+1$ over $\mathbb {Q}$, and $\mathbb {C}$ is the splitting field for the same polynomial, over $\mathbb {R}$.
     Example 4.4. The splitting field $F$ of $x^{8}-1$ over $\mathbb {Q}$ is generated by $\zeta :=2^{2\pi i/8}$; indeed, the roots of $x^{8}-1$ are all the 8-th roots of 1, and all of them are powers of $\zeta$: […] In fact, $\zeta$ is a root of the polynomial $x^{4}+1$, which is irreducible over $\mathbb {Q}$; therefore $F=\mathbb {Q} (\zeta )$ is 'already' the splitting field of $x^{4}+1$.

2. (algebra, ring theory, of a K-algebra) 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 $K\mapsto L$ such that $A\otimes _{K}L$ is split; in the special case $A=K[x]/f(x)$ 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 $T(R)=\operatorname {rad} R+[R,R]$, 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

     $T(R)\subseteq \{a\in R:a^{p^{m}}{\text{ for some }}m\geq 1\}$,

     with equality if $k$ is a splitting field for $R$.

   • 2016, Peter Webb, A Course in Finite Group Representation Theory, Cambridge University Press, page 162:

     Group algebras are defined over the prime field $\mathbb {Q}$ or $\mathbb {F} _{p}$ (depending on the characteristic), and by what we have just proved $\mathbb {Q} G$ and $\mathbb {F} _{p}G$ have splitting fields that are finite degree extensions of the prime field. […]
     Some other basic facts about splitting fields are left to the exercises at the end of this chapter. Thus, if $A$ is a finite-dimensional algebra over a field $F$ that is a splitting field for $A$ and $E\supset F$ is a field extension, it is the case that every simple $E\otimes _{F}A$-module can be written in $F$ (Exercises 4 and 8).

3. (algebra, ring theory, of a central simple algebra) 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.

   • 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 ${\mathfrak {C}}$ 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. ${\mathfrak {A}}$.

4. (algebra, 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.
   • 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,

     DEFINITION 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, $\mathbb {C}$ 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 $\chi \in \operatorname {Irr} (G)$.

Usage notes

• The polynomial (respectively, central simple algebra or character) is said to split over its splitting field.
• (Galois theory):
  • 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 normal extension of $K$.

Translations

(Galois theory, of a polynomial) smallest field containing all roots

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

(ring theory, of a K-algebra) extension field whose every simple module is absolutely simple

(ring theory, of a central simple algebra) field whose tensor product with the CSA is isomorphic to a matrix ring over said field

(character theory, of a group representation) field over which a K-representation exists that includes a given character

See also

• split K-algebra
• split Lie algebra

Further reading

• (Galois theory):
  • Splitting field of a polynomial on Encyclopedia of Mathematics
  • splitting field on nLab
  • Splitting Field on Wolfram MathWorld
• (ring theory: field on which a CSA splits):
  • Central simple algebra on Wikipedia.Wikipedia
  • Simple algebra on Wikipedia.Wikipedia
  • Tensor product of algebras on Wikipedia.Wikipedia
• (ring theory: field on winch a K-algebra splits):
  • Notes on Splitting Fields on University of Minnesota math user home pages (unreviewed)
  • What is a split K-algebra? on Mathematics Stack Exchange
• (character theory):
  • Character (mathematics) on Wikipedia.Wikipedia
  • character on nLab