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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( rac{2\pi}{17} ight)

is an example when it is written as

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

with

: \zeta = \exp \left( rac{2\pi 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} \exp\left( rac{2\pi 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) \exp\left( rac{2\pi 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 \exp\left( rac{2\pi imn}{k} ight) =


rac{-1+i\sqrt{p}}{2}, & \mbox{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


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 Equation s of L-function s. (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 Combination s of Gaussian periods, with coefficients χ(''a''); the converse is also true, as a consequence of the Orthogonality Relation s for the group (Z/''n''Z)×. In other words, the two sets of quantities are each other's Fourier Transform s. 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.