Eisenstein sum

Not to be confused with Eisenstein series.

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

${\displaystyle E(\chi ,\alpha )=\sum _{Tr_{F/K}t=\alpha }\chi (t)}$

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

1. (Eisenstein 1848)
2. (Stickelberger 1890)
3. (Yamamoto 1985)
4. (Lemmermeyer 2000, p. 133)

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, MR 0537798, Zbl 0393.12029
• 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
• Lemmermeyer, Franz (2000), Reciprocity laws, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66957-9, MR 1761696, Zbl 0949.11002
• Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields, Encyclopedia of Mathematics and Its Applications, vol. 20 (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4, Zbl 0866.11069
• Stickelberger, Ludwig (1890), "Ueber eine Verallgemeinerung der Kreistheilung", Mathematische Annalen, 37 (3): 321–367, doi:10.1007/bf01721360, JFM 22.0100.01, MR 1510649, S2CID 121239748
• 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, MR 0827799, Zbl 0634.12017