In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the area of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
, a Gaussian period is a certain kind of sum of
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
. The periods permit explicit calculations in
cyclotomic fields connected with
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
and with
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an ex ...
(
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a comple ...
). They are basic in the classical theory called cyclotomy. Closely related is the
Gauss sum, a type of
exponential sum which is a
linear combination of periods.
History
As the name suggests, the periods were introduced by
Gauss and were the basis for his theory of
compass and straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
construction. For example, the construction of the
heptadecagon (a formula that furthered his reputation) depended on the algebra of such periods, of which
:
is an example involving the seventeenth root of unity
:
General definition
Given an integer ''n'' > 1, let ''H'' be any
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the multiplicative group
:
of
invertible residues modulo ''n'', and let
:
A Gaussian period ''P'' is a sum of the
primitive n-th roots of unity
, where
runs through all of the elements in a fixed
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
of ''H'' in ''G''.
The definition of ''P'' can also be stated in terms of the
field trace
In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''.
Definition
Let ''K'' be a field and ''L'' a finite extension (and hence a ...
. We have
:
for some subfield ''L'' of Q(ζ) and some ''j'' coprime to ''n''. This corresponds to the previous definition by identifying ''G'' and ''H'' with the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
s of Q(ζ)/Q and Q(ζ)/''L'', respectively. The choice of ''j'' determines the choice of coset of ''H'' in ''G'' in the previous definition.
Example
The situation is simplest when ''n'' is a prime number ''p'' > 2. In that case ''G'' is cyclic of order ''p'' − 1, and has one subgroup ''H'' of order ''d'' for every factor ''d'' of ''p'' − 1. For example, we can take ''H'' of
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
two. In that case ''H'' consists of the
quadratic residue
In number theory, an integer ''q'' is called a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; i.e., if there exists an integer ''x'' such that:
:x^2\equiv q \pmod.
Otherwise, ''q'' is called a quadratic no ...
s modulo ''p''. Corresponding to this ''H'' we have the Gaussian period
:
summed over (''p'' − 1)/2 quadratic residues, and the other period ''P*'' summed over the (''p'' − 1)/2 quadratic non-residues. It is easy to see that
:
since the
left-hand side
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective al ...
, for a curve that is a
conic). One has
:(''P'' − ''P''*)
2 = ''p'' or −''p'', for ''p'' = 4''m'' + 1 or 4''m'' + 3 respectively.
This therefore gives us the precise information about which quadratic field lies in Q(ζ). (That could be derived also by
ramification arguments in
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
; see
quadratic field.)
As Gauss eventually showed, to evaluate ''P'' − ''P''*, the correct square root to take is the positive (resp. ''i'' times positive real) one, in the two cases. Thus the explicit value of the period ''P'' is given by
:
Gauss sums
As is discussed in more detail below, the Gaussian periods are closely related to another class of sums of roots of unity, now generally called Gauss sums (sometimes Gaussian sums). The quantity ''P'' − ''P''* presented above is a quadratic Gauss sum mod ''p'', the simplest non-trivial example of a Gauss sum. One observes that ''P'' − ''P''* may also be written as
:
where
here stands for the
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
(''a''/''p''), and the sum is taken over residue classes modulo ''p''. More generally, given a
Dirichlet character
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b:
:1) \ch ...
χ mod ''n'', the Gauss sum mod ''n'' associated with χ is
:
For the special case of
the
principal Dirichlet character, the Gauss sum reduces to the
Ramanujan sum:
:
where μ is the
Möbius function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most of ...
.
The Gauss sums
are ubiquitous in number theory; for example they occur significantly in the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
s of
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may giv ...
s. (Gauss sums are in a sense the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
analogues of the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
.)
Relationship of Gaussian periods and Gauss sums
The Gaussian periods are related to the Gauss sums
for which the character χ is trivial on ''H''. Such χ take the same value at all elements ''a'' in a fixed coset of ''H'' in ''G''. For example, the quadratic character mod ''p'' described above takes the value 1 at each quadratic residue, and takes the value -1 at each quadratic non-residue.
The Gauss sum
can thus be written as a linear combination of Gaussian periods (with coefficients χ(''a'')); the converse is also true, as a consequence of the
orthogonality relation
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information abou ...
s for the group (Z/''n''Z)
×. In other words, the Gaussian periods and Gauss sums are each other's
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s. The Gaussian periods generally lie in smaller fields, since for example when ''n'' is a prime ''p'', the values χ(''a'') are (''p'' − 1)-th roots of unity. On the other hand, Gauss sums have nicer algebraic properties.
References
*
{{Carl Friedrich Gauss
Galois theory
Cyclotomic fields
Euclidean plane geometry
Carl Friedrich Gauss