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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 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 .
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.
This means that the sum of two independent normally distributed random variables is normal, with its mean being the sum of the two means, and its variance being the sum of the two variances (i.e., the square of the standard deviation is the sum of the squares of the standard deviations). [1]
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reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Gauss, Carl Friedrich (1986), Disquisitiones Arithemeticae (Second, corrected edition), translated by Clarke, Arthur A.,
The standard Gaussian measure on . is a Borel measure (in fact, as remarked above, it is defined on the completion of the Borel sigma algebra, which is a finer structure);; is equivalent to Lebesgue measure: , where stands for absolute continuity of measures;
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 .
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