Gaussian period

In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. They permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis (discrete Fourier transform). They are basic in the classical theory called cyclotomy. Closely related are the Gauss sums, a type of exponential sum.

History

As the name suggests, they were introduced by Gauss 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

: $2\; cos\; left(frac\{2pi\}\{17\}\; ight)$

is an example when it is written as

: $zeta\; +\; zeta^\{16\}\; ,$

with

: $zeta\; =\; exp\; left(frac\{2pi\; i\}\{17\}\; ight).$

Basic sums

Gaussian periods have a rich theory. Some of the simplest results are that the summation:$g(n)\; =\; sum\_\{m=0\}^\{k-1\}\; expleft(frac\{2pi\; imn\}\{k\}\; ight)$is zero if "k" does not divide "n", and is equal to "k" if "k" divides "n". Given a Dirichlet character χ mod "k", the Gauss sum associated with χ is

:$G(n,chi)\; =\; sum\_\{m=1\}^k\; chi(m)\; expleft(frac\{2pi\; imn\}\{k\}\; ight)$

For the special case of $chi=chi\_1$ the principal Dirichlet character, the Gauss sum reduces to the Ramanujan sum:

:$G(n,chi\_1)\; =\; c\_k(n)\; =\; sum\_\{m=1;\; (m,k)=1\}^k\; expleft(frac\{2pi\; imn\}\{k\}\; ight)\; =sum\_\{d|(n,k)\}\; dmuleft(frac\{k\}\{d\}\; ight)$

where μ is the Möbius function.

General theory

In general, given an integer "n" > 1, the Gaussian periods are sums of various primitive n-th roots of 1, or in other words various sums of terms

: $zeta^a$

where

: $zeta\; =\; expleft(frac\{2pi\; i\}\{n\}\; ight)$

and "a" is an integer with ("a", "n") = 1. There is one such period "P" for each subgroup "H" of the group

: $G\; =\; (mathbb\{Z\}/nmathbb\{Z\})^\; imes$

of invertible residues modulo "n", and for each orbit "O" of "H" acting on the primitive "n"-th roots, by exponentiating. That is, we can make the definition

: $P\; =\; P(O)$

is the sum of the

: $zeta^a$

in the orbit "O".

Another form of this definition can be stated in terms of the field trace. We have

: $P\; =\; mathbf\{Tr\}\_\{mathbb\{Q\}(zeta)\; /\; L\}\; (zeta^j)$

for some subfield "L" of Q(ζ) and some "j" coprime to "n". Here to correspond to the previous form of definition one takes "H" to be the Galois group of Q(ζ)/"L", under the identification

: $mathbb\{Q\}(zeta)/mathbb\{Q\}\; =\; (mathbb\{Z\}/nmathbb\{Z\})^\; imes$

provided by choosing ζ as our reference root of unity.

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 two. In that case "H" consists of the quadratic residues modulo "p". Therefore an example of a Gaussian period is

: $P\; =\; zeta\; +\; zeta^4\; +\; zeta^9\; +\; cdots$

summed over ("p" − 1)/2 terms. There is also a period "P*" made up with exponents the quadratic non-residues. It is easy to see that we have

: $P\; +\; P^*\; =\; -1$

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. Squaring "P" as a sum leads to a counting problem, about how many quadratic residues are followed by quadratic residues, that can be solved by elementary methods (as we would now say, it computes a local zeta-function, for a curve that is a conic). This gives the result that

:("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; see quadratic field.)

As he eventually showed, the correct square root to take is the positive (resp. "i" times positive real) one, in the two cases. Thus the explicit value is given by

:

Gauss sums

The Gaussian periods are intimately related to another class of sums of roots of unity, now generally called Gauss sums (sometimes Gaussian sums). The quantity

:"P" − "P"*

that occurred above is the simplest non-trivial example. One observes that it may be written also

:$sum\; chi(a)zeta^a$

where χ("a") here stands for the Legendre symbol ("a"/"p"), and the sum is taken over residue classes modulo "p". The general case of Gauss sums replaces this choice for χ by any Dirichlet character modulo "n", the sum being taken over residue classes modulo "n" (with the usual convention that χ("a") = 0 if ("a","n") > 1).

These quantities are ubiquitous in number theory; for example they occur significantly in the functional equations of L-functions. (Gauss sums are in a sense the finite field analogues of the gamma function.)

Relationship of periods and sums

The relation with the Gaussian periods comes from the observation that the set of "a" modulo "n" at which χ("a") takes a given value is an orbit "O" of the type introduced earlier. Gauss sums can therefore be written as linear combinations of Gaussian periods, with coefficients χ("a"); the converse is also true, as a consequence of the orthogonality relations for the group (Z/"n"Z)^×. In other words, the two sets of quantities are each other's Fourier transforms. The Gaussian periods lie in smaller fields, in general, since the values of the χ("a") when "n" is a prime "p" are ("p" − 1)-th roots of unity. On the other hand the algebraic properties of Gauss sums are easier to handle.