User:Virginia-American/Sandbox/Gaussian Period

Given a positive integer n > 2, Gaussian periods are certain sums of the nth roots of unity. They were basic to Gauss's analysis of the cyclotomic polynomial, (now known as the classical theory of cyclotomy), and, more generally, permit explicit calculations in cyclotomic fields, in relation both with Galois theory and with harmonic analysis (discrete Fourier transform). Gauss sums, a type of exponential sum, are closely related.

Gauss's development
Gauss defines 'periods' in Section VII of the Disquisitiones Arithmeticae. In this section he is working under the assumption that the underlying integer n is a (positive odd) prime number.

This implies that there is a primitive root, g, (mod n). (That is, the powers g, g2, g3, ... gn &minus; 1 &equiv; 1 (mod n) are all distinct (mod n) and thus are the residue classes 1, 2, ..., n &minus; 1 (mod n) in a different order.)

Now let r be any primitive nth root of unity, (that is, rn = 1, but the powers r, r2, r3, ..., rn &minus; 1 are all distinct and none of them is equal to one). Now 1 is a root of xn &minus; 1 = 0, and dividing by x &minus; 1 gives


 * $$x^n - 1 = (x-1)(x^{n-1} + x^{n-2} + \cdots + x + 1)

$$

Gauss sets


 * $$X = x^{n-1} + x^{n-2} + \cdots + x + 1 $$ and defines $$\Omega\;$$ to be its set of zeroes.

Since X is of degree n &minus; 1, and the n &minus; 1 distinct numbers r, r2, r3, ..., rn &minus; 1are all zeroes of X,


 * $$X=(x-r)(x-r^2)\cdots(x-r^{n-1})

$$

and


 * $$\Omega = \{r, r^2, r^3, \cdots r^{n-1}\}.

$$ From the properties of the primitive root g, we also have


 * $$\Omega = \{r^g, r^{g^2}, r^{g^3}, \cdots r^{g^{n-1}}=r\},

$$

and in general, if &lambda; is not dvisible by n,


 * $$\Omega = \{r^{\lambda g}, r^{\lambda g^2}, r^{ \lambda g^3}, \cdots r^{\lambda g^{n-1}}=r^\lambda\}.

$$

Next, let e be any divisor of n &minus; 1:  $$n-1 = ef$$   and set   $$g^e=h,$$   so that   $$h^f\equiv 1 \pmod{n}.$$

Then, for any integer &lambda;, Gauss defines the period (f, &lambda;) of length f to be the set (multiset if n divides &lambda;) of f numbers


 * $$(f,\lambda) = \{r^\lambda, r^{\lambda h}, r^{\lambda h^2}, r^{ \lambda h^3}, \cdots r^{\lambda h^{f-1}}\}.

$$

Gauss notes that if  $$r^{\lambda '} \in (f,\lambda),$$   then   $$(f,\lambda) = (f,\lambda ')$$   and that   $$\Omega = (n-1, 1). $$ He states that   $$(f,\lambda) \subset \Omega $$   if &lambda; is not a multiple of n, but that    $$(f, 0) = (f, n)= (f,2n)\dots$$   is a multiset of f ones.

He shows that (f, &lambda;) is independent of the choice of primitive root.

He also notes that  $$\Omega$$   is a disjoint union of e periods of length f (where, as always, n = ef + 1):


 * $$\Omega = (f, \lambda)\cup (f, \lambda g) \cup (f, \lambda g^2) \cdots \cup (f, \lambda g^{e-1}).

$$

Modern treatments
Gauss's development of the theory of periods is a bit confusing at first, because he uses the same symbol for both the set and the sum of the numbers in the set. Most modern authors define the periods as sums and don't mention the sets at all. Using the same notation as in the preceding section (n = ef + 1 is an odd prime, g is a primitive root (mod n), r is a primitive nth root of unity, &lambda; an arbitrary integer), some authors simply define the period of length f as the sum:


 * $$(f,\lambda) = \sum_{j=0}^{f-1}r^{\lambda g^{ej}}.

$$

Others employ a different notation. Let $$\sigma$$ denote the automorphism of the cyclotomic field $$\mathbb{Q}(r)$$ that sends $$r$$ to $$r^g,$$ (this is called a conjugation), and use exponential notation to denote iterating it:

$$\sigma^2(r) = r^{g^2}, \sigma^3(r) = r^{g^3},$$  etc. Note that $$\sigma^n$$ is the identity, and that the exponents can be taken (mod n).

Then, given e and f as above, define the e periods of length f as:


 * $$\begin{align}

\eta_0 &= r+\sigma^er + \sigma^{2e}r + \sigma^{3e}r+\cdots+ \sigma^{(f-1)e}r \\ \eta_1 &= \sigma\eta_0\\ \eta_2 &= \sigma\eta_1\\ &\vdots\\ \eta_{e-1} &= \sigma\eta_{e-2} \end{align} $$

Note that the subscripts may be taken (mod e), and that unlike the periods denoted by (f, &lambda;), the names (not the values) of the periods depend on the primitive root. (i.e., picking a different primitive root may permute some of the &eta;i s.)

Let $$\nu(a)$$ denote the index of a (mod n) relative to the primitive root g (sometimes called the discrete logarithm):
 * $$g^{\nu(a)}\equiv a \pmod{n}.

$$ Then the above construction can be summarized as


 * $$\eta_j = \sum_{\stackrel{0\le k < n}{\nu(k)\equiv j \pmod{e}}}r^k.

$$

n = 3, g = 2
There are two periods of length 1



\begin{align} \eta_0 &= r \\ \eta_1 &= r^2 = r^{-1} \\ \end{align} $$

and one period of length 2


 * $$\eta_0 = r + r^2.\;

$$

n = 5, g = 2
There are 4 periods of length 1



\begin{align} \eta_0 &= r \\ \eta_1 &= r^2 \\ \eta_2 &= r^4 \\ \eta_3 &= r^8 = r^3, \end{align} $$

2 periods of length 2



\begin{align} \eta_0 &= r + r^4\\ \eta_1 &= r^2+ r^3, \end{align} $$

and 1 period of length 4


 * $$\eta_0 = r + r^2 + r^4 + r^3.\;

$$

n = 7, g = 3
There are 6 periods of length 1



\begin{align} \eta_0 &= r \\ \eta_1 &= r^3 \\ \eta_2 &= r^9 &=&\;\; r^2 \\ \eta_3 &= r^{27} &=&\;\;r^6\\ \eta_4 &= r^{81}&=&\;\;r^4\\ \eta_5 &= r^{243} &=&\;\; r^5, \end{align} $$

3 periods of length 2



\begin{align} \eta_0 &= r + r^{27} & = & \;\;r+r^6\\ \eta_1 &= r^3 + r^{81} & = & \;\;r^3+r^4\\ \eta_2 &= r^9+ r^{243} & = & \;\;r^2+r^5, \end{align} $$

2 periods of length 3



\begin{align} \eta_0 &= r + r^{9} + r^{81} & = & \;\; r+r^2+ r^4\\ \eta_1 &= r^3+ r^{27} + r^{243}& = & \;\; r^3+r^6+r^5, \end{align} $$

and 1 period of length 6


 * $$\eta_0 = r + r^3 + r^2 + r^6 + r^4 + r^5.\;

$$

n = 11, g = 2
There are 10 periods of length 1



\begin{align} \eta_0 &= r \\ \eta_1 &= r^2 \\ \eta_2 &= r^4 \\ \eta_3 &= r^{8} \\ \eta_4 &= r^{16}&=&\;\;r^5\\ \eta_5 &= r^{32}&=&\;\;r^{10} \\ \eta_6 &= r^{64} &=&\;\; r^9 \\ \eta_7 &= r^{128}& =&\;\;r^7\\ \eta_8 &= r^{256}&=&\;\;r^3\\ \eta_9 &= r^{512} &= &\;\;r^6, \end{align} $$

5 periods of length 2



\begin{align} \eta_0 &= r + r^{32} & = & \;\;r+r^{10}\\ \eta_1 &= r^2 + r^{64} & = & \;\;r^2+r^9\\ \eta_2 &= r^4 + r^{128} & = & \;\;r^4+r^7\\ \eta_3 &= r^8 + r^{256} & = & \;\;r^8+r^3\\ \eta_4 &= r^{16}+ r^{512} & = & \;\;r^5+r^6, \end{align} $$

2 periods of length 5



\begin{align} \eta_0 &= r + r^{4} + r^{16} + r^{64} + r^{256}& = & \;\; r+r^4+ r^5+ r^9+r^3\\ \eta_1 &= r^2+ r^{8} + r^{32} + r^{128} + r^{512}&= & \;\; r^2+r^8+r^{10}+ r^7 + r^6, \end{align} $$

and 1 period of length 10


 * $$\eta_0 = r + +r^2 + r^4 + r^8 + r^5 + r^{10} + r^9 + r^7 + r^3 + r^6.\;

$$

n = 13, g = 2
There are 12 periods of length 1



\begin{align} \eta_0   &= r \\ \eta_1   &= r^2 \\ \eta_2   &= r^4  \\ \eta_3   &= r^{8} \\ \eta_4   &= r^{16}   & = & \;\; r^3\\ \eta_5   &= r^{32}   & = & \;\; r^{6} \\ \eta_6   &= r^{64}   & = & \;\; r^{12} \\ \eta_7   &= r^{128}  & = & \;\; r^{11}\\ \eta_8   &= r^{256}  & = & \;\; r^9\\ \eta_9   &= r^{512}  & = & \;\; r^5\\ \eta_{10} &= r^{1024} & = & \;\; r^{10}\\ \eta_{11} &= r^{2048} & = & \;\; r^7, \end{align} $$

6 periods of length 2



\begin{align} \eta_0 &= r + r^{64} & = & \;\;r+r^{12}\\ \eta_1 &= r^2 + r^{128} & = & \;\;r^2+r^{11}\\ \eta_2 &= r^4 + r^{256} & = & \;\;r^4+r^9\\ \eta_3 &= r^8 + r^{512} & = & \;\;r^8+r^5\\ \eta_4 &= r^{16} + r^{1024} & = & \;\;r^3+r^{10}\\ \eta_5 &= r^{32}+ r^{2048} & = & \;\;r^6+r^7, \end{align} $$

4 periods of length 3



\begin{align} \eta_0 &= r + r^{16} + r^{256} & = & \;\;r+r^{3}+r^{9}\\ \eta_1 &= r^2 + r^{32} + r^{512} & = & \;\;r^2+r^{6}+r^{5}\\ \eta_2 &= r^4 + r^{64} + r^{1024} & = & \;\;r^4+r^{12}+r^{10}\\ \eta_3 &= r^{8}+ r^{128} + r^{2048} & = & \;\;r^8+r^{11}+r^{7}, \end{align} $$

3 periods of length 4



\begin{align} \eta_0 &= r + r^{8} + r^{64} + r^{512} & = & \;\;r+r^{8}+r^{12}+ r^{5} \\ \eta_1 &= r^2 + r^{16} + r^{128} + r^{1024} & = & \;\;r^2+r^{3}+r^{11}+ r^{10} \\ \eta_2 &= r^{4}+ r^{32} + r^{256} + r^{2048} & = & \;\;r^4+r^{6}+r^{9}+ r^{7} , \end{align} $$

2 periods of length 6



\begin{align} \eta_0 &= r + r^{4} + r^{16} + r^{64} + r^{256} + r^{1024} & = & \;\; r+r^4+ r^3+ r^{12}+r^9+ r^{10} \\ \eta_1 &= r^2+ r^{8} + r^{32} + r^{128} + r^{512} + r^{2048} &=& \;\; r^2+r^8+r^{6}+ r^{11} + r^5+ r^{7} , \end{align} $$

and 1 period of length 12


 * $$\eta_0 = r + +r^2 + r^4 + r^8 + r^3 + r^{6} + r^{12} + r^{11} + r^9 + r^5 + r^{10} + r^7.\;

$$

Every period is a sum of shorter periods
If the length f = a b, a period of length f is the sum of b periods of length a, or of a periods of length b.

Denote the length of a period with a superscript, e.g. for n = 13,


 * $$\eta_0^{(1)} = r,\;\;\;\;\eta_0^{(6)}=r+r^4+r^9+r^{16}+r^{25}+r^{36}.

$$

Then for example, for n = 13,


 * $$\eta_0^{(6)} = \eta_0^{(3)} +\eta_2^{(3)}.

$$

In fact, the periods may be arranged in a tree, where the one on the left is the sum of those to the right:



\eta_{0}^{(12)} = \begin{cases} \eta_{0}^{(6)} = \begin{cases} \eta_{0}^{(3)} = \begin{cases} \eta_{0}^{(1)}=r^{} \\ \eta_{4}^{(1)}=r^{3} \\ \eta_{8}^{(1)}=r^{9} \end{cases}\\ \\ \eta_{2}^{(3)} = \begin{cases} \eta_{2}^{(1)}=r^{4} \\ \eta_{6}^{(1)}=r^{12} \\ \eta_{10}^{(1)}=r^{10} \end{cases}\\ \end{cases}\\ \;\\ \eta_{1}^{(6)} = \begin{cases} \eta_{1}^{(3)} = \begin{cases} \eta_{1}^{(1)}=r^{2} \\ \eta_{5}^{(1)}=r^{6} \\ \eta_{9}^{(1)}=r^{5} \end{cases}\\\\ \eta_{3}^{(3)} = \begin{cases} \eta_{3}^{(1)}=r^{8} \\ \eta_{7}^{(1)}=r^{11} \\ \eta_{11}^{(1)}=r^{7} \end{cases}\\ \end{cases}\\ \end{cases}

\mbox{ or }\;\;

\eta_{0}^{(12)} = \begin{cases} \eta_{0}^{(4)} = \begin{cases} \eta_{0}^{(2)} = \begin{cases} \eta_{0}^{(1)}=r^{} \\ \eta_{6}^{(1)}=r^{12} \end{cases}\\ \\ \eta_{3}^{(2)} = \begin{cases} \eta_{3}^{(1)}=r^{8} \\ \eta_{9}^{(1)}=r^{5} \end{cases}\\ \end{cases}\\ \;\\ \eta_{1}^{(4)} = \begin{cases} \eta_{1}^{(2)} = \begin{cases} \eta_{1}^{(1)}=r^{2} \\ \eta_{7}^{(1)}=r^{11} \end{cases}\\ \\ \eta_{4}^{(2)} = \begin{cases} \eta_{4}^{(1)}=r^{3} \\ \eta_{10}^{(1)}=r^{10} \end{cases}\\ \end{cases}\\ \;\\ \eta_{2}^{(4)} = \begin{cases} \eta_{2}^{(2)} = \begin{cases} \eta_{2}^{(1)}=r^{4} \\ \eta_{8}^{(1)}=r^{9} \end{cases}\\\\ \eta_{5}^{(2)} = \begin{cases} \eta_{5}^{(1)}=r^{6} \\ \eta_{11}^{(1)}=r^{7} \end{cases}\\ \end{cases}\\ \end{cases} $$

Invariance under certain automorphisms
Any polynomial in an nth root of unity r may be reduced to one hose degree is less than n by using the identities $$r^n =1,\;\;r^{n+1}= r, \;\;r^{n+2}= r^2, \dots$$

Product of two periods
The product of two periods of length f is a linear expression in the perids of length f with integer coefficients.

Period polynomial
The periods of length f are the roots of an eth degree polynomial with integer coefficients. This polynomial is irrdeucible over the rationals.

For any n, the n &minus; 1 periods of length 1 are the roots of the cyclotomic polynomial
 * $$x^{n-1} + x^{n-2} + \cdots + x + 1.

$$

Similarly, for any n, the period of length n &minus; 1 is &minus;1.

Gaussian periods have a rich theory. Some of the simplest results are that the summation
 * $$g(n) = \sum_{m=0}^{k-1} \exp\left(\frac{2\pi imn}{k}\right) $$

is zero if k does not divide n, and is equal to k if k divides n. Given a Dirichlet character &chi; mod k, the Gauss sum associated with &chi; is


 * $$G(n,\chi) = \sum_{m=1}^k \chi(m) \exp\left(\frac{2\pi imn}{k}\right). $$

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 \exp\left(\frac{2\pi imn}{k}\right) = \sum_{d|(n,k)} d\mu\left(\frac{k}{d}\right) $$

where &mu; 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


 * $$ \zeta = \exp\left(\frac{2\pi i}{n}\right) $$

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


 * $$ G = (\mathbb{Z}/n\mathbb{Z})^\times $$

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(&zeta;) 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(&zeta;)/L, under the identification


 * $$ \mathbb{Q}(\zeta)/\mathbb{Q} = (\mathbb{Z}/n\mathbb{Z})^\times $$

provided by choosing &zeta; 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 &minus; 1, and has one subgroup H of order d for every factor d of p &minus; 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 &minus; 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 &minus; P*)2 = p or &minus;p, for p = 4m + 1 or 4m + 3 respectively.

This therefore gives us the precise information about which quadratic field lies in Q(&zeta;). (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


 * $$ P = \begin{cases} \frac{-1+\sqrt{p}}{2}, & \text{if }p=4m+1, \\[6pt]

\frac{-1+i\sqrt{p}}{2}, & \text{if }p=4m+3. \end{cases}$$

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 &minus; P*

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


 * $$\sum \chi(a)\zeta^a$$

where &chi;(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 &chi; by any Dirichlet character modulo n, the sum being taken over residue classes modulo n (with the usual convention that &chi;(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 &chi;(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 &chi;(a); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)&times;. 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 &chi;(a) when n is a prime p are (p &minus; 1)-th roots of unity. On the other hand the algebraic properties of Gauss sums are easier to handle.