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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 ...
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.
It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
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 .
The German edition includes all of Gauss's 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reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n". Gauss, Carl Friedrich (1986).
Gauss originally used the Gaussian binomial coefficients in his determination of the sign of the quadratic Gauss sum. [3] Gaussian binomial coefficients occur in the counting of symmetric polynomials and in the theory of partitions. The coefficient of q r in (+)
However, Gauss's law can be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the superposition principle. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).