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In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically ():= (,) = ()where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R + into the unit circle, and χ is a group homomorphism of the unit group R × into the unit circle, extended to non-unit r, where it takes the ...
In mathematics, the Gross–Koblitz formula, introduced by Gross and Koblitz expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem.
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 transforms.
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum .
This is an accepted version of this page This is the latest accepted revision, reviewed on 11 February 2025. German mathematician, astronomer, geodesist, and physicist (1777–1855) "Gauss" redirects here. For other uses, see Gauss (disambiguation). Carl Friedrich Gauss Portrait by Christian Albrecht Jensen, 1840 (copy from Gottlieb Biermann, 1887) Born Johann Carl Friedrich Gauss (1777-04-30 ...
Gauss also writes, "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work." ("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri ...
The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
A fundamental property of these Gauss sums is that = where = (). To put this in context of the next proof, the individual elements of the Gauss sum are in the cyclotomic field L = Q ( ζ p ) {\displaystyle L=\mathbb {Q} (\zeta _{p})} but the above formula shows that the sum itself is a generator of the unique quadratic field contained in L .