Note on the quadratic Gauss sums

George Danas

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 25, 1-7

Abstract:

Let p be an odd prime and { χ ( m ) = ( m / p ) } , m = 0 , 1 , ... , p − 1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ mod p which are defined in terms of the Legendre symbol ( m / p ) , ( m , p ) = 1 . We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sums G ( k ; p ) are equal to the Gauss sums G ( k , χ ) that correspond to this particular Dirichlet character χ . Finally, using the above result, we prove that the quadratic Gauss sums G ( k ; p ) , k = 0 , 1 , ... , p − 1 are the eigenvalues of the circulant p × p matrix X with elements the terms of the sequence { χ ( m ) } .

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:345676

DOI: 10.1155/S016117120100480X

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