primitive element

Fetched 2026-10-26 (cache).

Datamuse reports no synonyms or antonyms for this word.

Noun

1. {{lb|en|algebra|field theory}} An [[element]] that [[generate]]s a [[simple extension]].
2. {{lb|en|algebra|field theory|of a finite field}} An [[element]] that [[generate]]s the [[multiplicative]] [[group]] of a given [[Galois field]] (finite [[field]]).
3. {{lb|en|number theory}} Given a [[modulus]] ''n'', a number ''g'' such that every number [[coprime]] to ''n'' is [[congruent]] ([[modulo]] ''n'') to some power of ''g''; ''equivalently'', a [[generator]] of the [[multiplicative field]] of integers modulo ''n''.
4. {{lb|en|algebra|lattice theory|of a lattice}} An [[element]] that is not a positive integer multiple of another element of the [[lattice]].
5. {{lb|en|algebra|of a coalgebra over an element g}} An [[element]] ''x'' ∈ ''C'' such that ''μ''(''x'') = ''x'' ⊗ ''g + g'' ⊗ ''x'', where ''μ'' is the [[comultiplication]] and ''g'' is an element that maps to the [[multiplicative identity]] 1 of the base field under the [[counit]] (in particular, if ''C'' is a [[bialgebra]], ''g'' = 1).
6. {{lb|en|group theory|of a [[free group]]}} An element of a [[free generating set]] of a given free group.

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