Eisenstein sum
From HandWiki
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 Eisenstein in 1848,[1] named "Eisenstein sums" by Stickelberger in 1890,[2] and rediscovered by Yamamoto in 1985,[3] who called them relative Gauss sums.
Definition
The Eisenstein sum is given by
- [math]\displaystyle{ E(\chi,\alpha)=\sum_{Tr_{F/K}t=\alpha}\chi(t) }[/math]
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.[4]
References
Bibliography
- Berndt, Bruce C.; Evans, Ronald J. (1979), "Sums of Gauss, Eisenstein, Jacobi, Jacobsthal, and Brewer", Illinois Journal of Mathematics 23 (3): 374–437, doi:10.1215/ijm/1256048104, ISSN 0019-2082, http://projecteuclid.org/getRecord?id=euclid.ijm/1256048104
- Eisenstein, Gotthold (1848), "Zur Theorie der quadratischen Zerfällung der Primzahlen 8n + 3,7n + 2 und 7n + 4", Journal für die Reine und Angewandte Mathematik 37: 97–126, ISSN 0075-4102, http://resolver.sub.uni-goettingen.de/purl?GDZPPN002146460
- Lemmermeyer, Franz (2000), Reciprocity laws, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66957-9, https://books.google.com/books?id=EwjpPeK6GpEC
- Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields, Encyclopedia of Mathematics and Its Applications, 20 (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4, https://archive.org/details/finitefields0000lidl_a8r3
- Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen 37 (3): 321–367, doi:10.1007/bf01721360, http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=27547
- Yamamoto, K. (1985), "On congruences arising from relative Gauss sums", Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984), Singapore: World Sci. Publishing, pp. 423–446
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