https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Whitham_equationWhitham equation - Revision history2026-09-20T20:28:40ZRevision history for this page on the wikiMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Whitham_equation&diff=28150&oldid=preven>David Eppstein: {{applied-math-stub}}2013-10-04T18:07:47Z<p>{{applied-math-stub}}</p>
<p><b>New page</b></p><div>{{distinguish|Eisenstein series}}<br />
In mathematics, an '''Eisenstein sum''' is a finite sum depending on a finite field and related to a [[Gauss sum]]. Eisenstein sums were introduced by {{harvs|txt|last=Eisenstein|first=Gotthold|year=1848}}, named "Eisenstein sums" by {{harvtxt|Stickelberger|1890}}, and rediscovered by {{harvtxt|Yamamoto|1985}}, who called them '''relative Gauss sums'''.<br />
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==Definition==<br />
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The Eisenstein sum is given by<br />
:<math>E(\chi,\alpha)=\sum_{Tr_{F/K}t=\alpha}\chi(t)</math><br />
where ''F'' is a finite extension of the finite field ''K'', and χ is a character of the multiplicative group of ''F'', and α is an element of ''K'' {{harv|Lemmermeyer|2000|loc=p. 133}}.<br />
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==References==<br />
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*{{Citation | last1=Berndt | first1=Bruce C. | last2=Evans | first2=Ronald J. | title=Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer | url=http://projecteuclid.org/getRecord?id=euclid.ijm/1256048104 | mr=537798 | zbl=0393.12029 | year=1979 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=23 | issue=3 | pages=374–437}}<br />
*{{Citation | last1=Eisenstein | first1=Gotthold | title=Zur Theorie der quadratischen Zerfällung der Primzahlen 8n + 3,7n + 2 und 7n + 4 | year=1848 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=37 | pages=97–126 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002146460}}<br />
*{{Citation | last1=Lemmermeyer | first1=Franz | title=Reciprocity laws | url=http://books.google.com/books?id=EwjpPeK6GpEC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-66957-9 | mr=1761696| zbl=0949.11002 | year=2000}}<br />
*{{citation | zbl=0866.11069 | last1=Lidl | first1=Rudolf | last2=Niederreiter | first2=Harald | title=Finite fields | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=20 | publisher=[[Cambridge University Press]] | year=1997 | isbn=0-521-39231-4 }} <br />
*{{Citation | last1=Yamamoto | first1=K. | title=Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) | publisher=World Sci. Publishing | location=Singapore | mr=827799 | zbl=0634.12017 | year=1985 | chapter=On congruences arising from relative Gauss sums | pages=423–446}}<br />
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[[Category:Number theory]]</div>en>David Eppstein