Gaussian period
Encyclopedia
In mathematics
, in the area of number theory
, a Gaussian period is a certain kind of sum of roots of unity
. The periods permit explicit calculations in cyclotomic field
s connected with Galois theory
and with harmonic analysis
(discrete Fourier transform
). 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.
and were the basis for his theory of compass and straightedge
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
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
of H in G.
The definition of P can also be stated in terms of the field trace
. 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
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.
two. In that case H consists of the quadratic residues 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 LHS
adds all the primitive p-th roots of 1. We also know, from the trace definition, that P lies in a quadratic extension of Q. Therefore, as Gauss knew, P satisfies a quadratic equation with integer coefficients. Evaluating the square of the sum P is connected with the problem of counting how many quadratic residues between 1 and p − 1 are succeeded by quadratic residues. The solution is elementary (as we would now say, it computes a local zeta-function
, for a curve that is a conic). One has
2 = p or −p, for p = 4m + 1 or 4m + 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
; 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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the area of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a Gaussian period is a certain kind of sum of roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
. The periods permit explicit calculations in cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s connected with Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
and with harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
(discrete Fourier transform
Discrete Fourier transform
In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
). They are basic in the classical theory called cyclotomy. Closely related is the Gauss sum
Gauss sum
In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...
, a type of exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...
which is a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of periods.
History
As the name suggests, the periods were introduced by GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
and were the basis for his theory of compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
construction. For example, the construction of the heptadecagon
Heptadecagon
In geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
(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 subgroupSubgroup
In group theory, 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 G if the restriction of * to H x H is a group operation on H...
of the multiplicative group
of invertible residues
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
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 cosetCoset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
of H in G.
The definition of P can also be stated in terms of the field trace
Field trace
In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K...
. 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
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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 indexIndex of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
two. In that case H consists of the quadratic residues 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 LHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
adds all the primitive p-th roots of 1. We also know, from the trace definition, that P lies in a quadratic extension of Q. Therefore, as Gauss knew, P satisfies a quadratic equation with integer coefficients. Evaluating the square of the sum P is connected with the problem of counting how many quadratic residues between 1 and p − 1 are succeeded by quadratic residues. The solution is elementary (as we would now say, it computes a local zeta-function
Local zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
, for a curve that is a conic). One has
2 = p or −p, for p = 4m + 1 or 4m + 3 respectively.
This therefore gives us the precise information about which quadratic field lies in Q(ζ). (That could be derived also by ramification
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...
arguments in algebraic number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
; see quadratic field
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
.)
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 symbolLegendre symbolIn number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo a prime number p: its value on a quadratic residue mod p is 1 and on a quadratic non-residue is −1....
(a/p), and the sum is taken over residue classes modulo p. More generally, given a Dirichlet characterDirichlet characterIn number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
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 functionMöbius functionThe classical Möbius function μ is an important multiplicative function in number theory and combinatorics. The German mathematician August Ferdinand Möbius introduced it in 1832...
.
The Gauss sums
are ubiquitous in number theory; for example they occur significantly in the functional equationFunctional equationIn mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
s of L-functionL-functionThe theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s. (Gauss sums are in a sense the finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
analogues of the gamma functionGamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.)
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 relations for the group (Z/nZ)×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
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.