Phrase guide

primitive element

noun · 6 senses · updated from the 2026-07-25 local source snapshot

Definitions and examples are grouped by meaning. Pronunciation, history, word forms, translations, descendants, synonyms, antonyms, derived terms, and related words appear whenever the source provides them.

Sound

Pronunciation

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1

noun

Meaning 1

An element that generates a simple extension.

Definition source: English Wiktionary via Wiktextract

Qualifier: field theory

Topics: algebra, mathematics, sciences

No example sentence was provided for this meaning.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

2

noun

Meaning 2

An element that generates the multiplicative group of a given Galois field (finite field).

Definition source: English Wiktionary via Wiktextract

Qualifier: field theory

Topics: algebra, mathematics, sciences

Examples

  • 2003, Soonhak Kwon, Chang Hoon Kim, Chun Pyu Hong, Efficient Exponentiation for a Class of Finite Fields GF(2n) Determined by Gauss Periods, Colin D. Walter, Çetin K. Koç, Christof Paar (editors), Cryptographic Hardware and Embedded Systems, CHES 2003: 5th International Workshop, Proceedings, Springer, LNCS 2779, page 228, Also in the case of a Gauss period of type (n,1), i.e. a type I optimal normal element, we find a primitive element in GF(2n) which is a sparse polynomial of a type I optimal normal element and we propose a fast exponentiation algorithm which is applicable for both software and hardware purposes.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

3

noun

Meaning 3

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.

Definition source: English Wiktionary via Wiktextract

Topics: mathematics, number-theory, sciences

No example sentence was provided for this meaning.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

4

noun

Meaning 4

An element that is not a positive integer multiple of another element of the lattice.

Definition source: English Wiktionary via Wiktextract

Qualifier: lattice theory

Topics: algebra, mathematics, sciences

No example sentence was provided for this meaning.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

5

noun

Meaning 5

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).

Definition source: English Wiktionary via Wiktextract

Topics: algebra, mathematics, sciences

Examples

  • 2009, Masoud Khalkhali, Basic Noncommutative Geometry, European Mathematical Society, page 29, A primitive element of a Hopf algebra is an element h∈H such that Δh=1⊗h+h⊗1. It is easily seen that the bracket [x,y]:=xy-yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For H=U(g) any element of g is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

6

noun

Meaning 6

An element of a free generating set of a given free group.

Definition source: English Wiktionary via Wiktextract

Topics: group-theory, mathematics, sciences

No example sentence was provided for this meaning.

Meaning relationships

Synonyms: none provided

Antonyms: none provided

History

Etymology

No etymology was provided for this word.

Across languages

Translations

8 source translations are retained for this English entry.

  • Finnish: primitiivinen alkio — element that generates a field extension
  • Polish: element pierwotny — element that generates a field extension
  • Finnish: primitiivinen alkio — element of a finite field that generates its multiplicative group
  • Polish: element pierwotny — element of a finite field that generates its multiplicative group
  • Finnish: primitiivinen alkio — element of a lattice that is not a positive multiple of another element
  • Polish: element pierwotny — element of a lattice that is not a positive multiple of another element
  • Finnish: primitiivinen alkio — element of a coalgebra satisfying a particular condition
  • Polish: element pierwotny — element of a coalgebra satisfying a particular condition