Gaussian periods

E157384

mathematical concept number theoretic object

Gaussian periods are special algebraic sums of roots of unity that play a key role in number theory, particularly in constructing regular polygons like the 17-gon with straightedge and compass and in understanding cyclotomic fields.

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All labels observed (2)

Label	Occurrences
Gaussian periods canonical	1
Kummer sums	1

Statements (46)

Predicate	Object
instanceOf	mathematical concept ⓘ number theoretic object ⓘ
appearsIn	Disquisitiones Arithmeticae ⓘ
definedAs	sums of roots of unity over cosets of a subgroup of the multiplicative group modulo a prime ⓘ
developedBy	Carl Friedrich Gauss ⓘ
field	number theory ⓘ
generalizationOf	quadratic Gauss sums ⓘ
hasProperty	algebraic integers ⓘ closed under Galois conjugation ⓘ generate subfields of cyclotomic fields ⓘ linearly independent over the rationals in typical constructions ⓘ satisfy polynomial equations with integer coefficients ⓘ their conjugates are obtained by multiplying indices by units modulo the defining modulus ⓘ their minimal polynomials often have relatively small coefficients ⓘ
namedAfter	Carl Friedrich Gauss ⓘ
parameterizedBy	a modulus, typically a prime p ⓘ a subgroup of the multiplicative group modulo the modulus ⓘ cosets of that subgroup ⓘ
relatedTo	Dirichlet characters ⓘ Galois theory ⓘ Gauss periods in coding theory ⓘ Gaussian periods of type (k,n) ⓘ Hilbert class field ⓘ surface form: Hilbert class fields of imaginary quadratic fields Kummer theory ⓘ Weber functions and modular invariants in some class field constructions ⓘ constructible polygons ⓘ cyclotomic fields ⓘ cyclotomic polynomials ⓘ cyclotomic units ⓘ finite fields ⓘ regular 17-gon ⓘ roots of unity ⓘ
specialCaseOf	periods in algebraic number theory ⓘ
usedFor	analyzing distribution of residues modulo primes ⓘ constructing certain cyclic difference sets ⓘ constructing normal bases ⓘ constructing normal bases of finite field extensions ⓘ describing subfields of cyclotomic fields ⓘ evaluating Gauss sums ⓘ explicit class field theory over Q ⓘ explicit construction of regular polygons ⓘ explicit description of intermediate fields in cyclotomic extensions ⓘ explicit generators of abelian extensions of Q ⓘ straightedge and compass constructions ⓘ studying higher power residues ⓘ studying quadratic residues ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

construction of the regular 17-gon with straightedge and compass → isLinkedTo → Gaussian periods ⓘ

Ernst Eduard Kummer → knownFor → Gaussian periods ⓘ

this entity surface form: Kummer sums