Each Play button uses your device's default local English voice, or a natural local fallback. The selected voice name appears after playback. This is synthesized speech, not a source recording.
No pronunciation record was provided for this word.
(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.
Definition source: English Wiktionary via Wiktextract
Qualifier: Galois theory
Topics: algebra, mathematics, sciences
No example sentence was provided for this meaning.
Meaning relationships
Synonyms: root field
Antonyms: none provided
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).
Definition source: English Wiktionary via Wiktextract
Qualifier: ring theory
Topics: algebra, mathematics, sciences
Examples
- 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.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
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.
Definition source: English Wiktionary via Wiktextract
Qualifier: ring theory
Topics: algebra, mathematics, sciences
Examples
- 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.
Meaning relationships
Synonyms: none provided
Antonyms: none provided
(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 source: English Wiktionary via Wiktextract
Qualifier: character theory
Topics: algebra, mathematics, sciences
Examples
- 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 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 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 χ1,χ2, then K is a splitting field of χ1+χ2, 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 χ∈ operatorname Irr(G).
Meaning relationships
Synonyms: none provided
Antonyms: none provided
No etymology was provided for this word.
Across languages
Translations
15 source translations are retained for this English entry.
-
Basque:
deskonposizio gorputz
— (Galois theory, of a polynomial) smallest field containing all roots
-
Basque:
banatze gorputz
— (Galois theory, of a polynomial) smallest field containing all roots
-
French:
corps de décomposition
— (Galois theory, of a polynomial) smallest field containing all roots
-
French:
corps des racines
— (Galois theory, of a polynomial) smallest field containing all roots
-
French:
corps de déploiement
— (Galois theory, of a polynomial) smallest field containing all roots
-
German:
Zerfällungskörper
— (Galois theory, of a polynomial) smallest field containing all roots
-
Italian:
campo di spezzamento
— (Galois theory, of a polynomial) smallest field containing all roots
-
Italian:
campo di riducibilità completa
— (Galois theory, of a polynomial) smallest field containing all roots
-
Polish:
pole podziału
— (Galois theory, of a polynomial) smallest field containing all roots
-
Portuguese:
corpo de decomposição
— (Galois theory, of a polynomial) smallest field containing all roots
-
Portuguese:
corpo de fatoração
— (Galois theory, of a polynomial) smallest field containing all roots
-
Spanish:
cuerpo de descomposición
— (Galois theory, of a polynomial) smallest field containing all roots
-
Polish:
pole podziału
— (ring theory, of a K-algebra) extension field whose every simple module is absolutely simple
-
Polish:
pole podziału
— (ring theory, of a central simple algebra) field whose tensor product with the CSA is isomorphic to a matrix ring over said field
-
Polish:
pole podziału
— (character theory, of a group representation) field over which a K-representation exists that includes a given character
