splitting field

Fetched 2026-10-26 (cache).

Datamuse reports no synonyms or antonyms for this word.

Noun

1. {{lb|en|algebra|Galois theory}} {{lb|en|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]] [[factor]]s (polynomials of degree 1); {{lb|en|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.
2. {{lb|en|algebra|ring theory|of a K-algebra}} Given a finite-dimensional [[K-algebra|''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. {{lb|en|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''.
4. {{lb|en|algebra|[[character theory]]}} {{lb|en|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; {{lb|en|of a group G}} a field over which a ''K''-representation of ''G'' exists which includes every [[irreducible]] character in ''G''.

Source: Wiktionary · CC BY-SA 4.0 (Wiktionary dump) · cached copy, not a rewrite.