On the evaluation of Brewer’s character sums

Giudici, Reinaldo E.; Muskat, Joseph B.; Robinson, Stanley F.

doi:10.1090/S0002-9947-1972-0306122-5

On the evaluation of Brewer’s character sums
HTML articles powered by AMS MathViewer

by Reinaldo E. Giudici, Joseph B. Muskat and Stanley F. Robinson PDF

Trans. Amer. Math. Soc. 171 (1972), 317-347 Request permission

Abstract:

A decade ago in this journal B. W. Brewer defined a sequence of polynomials ${V_n}(x,1)$ and for $n = 4$ and 5 evaluated \[ \sum \limits _{x = 1}^p {{}_\chi ({V_n}(x,1))}, \] $\chi$ the nonprincipal quadratic character of the prime $p$, in closed form. A. L Whiteman derived these results by means of cyclotomy. Brewer subsequently defined ${V_n}(x,Q)$. This paper applies cyclotomy to the more general polynomials and provides evaluations for several more values of $n$. Relevant quadratic decompositions of primes are studied.

References

B. W. Brewer, On certain character sums, Trans. Amer. Math. Soc. 99 (1961), 241–245. MR 120202, DOI 10.1090/S0002-9947-1961-0120202-1
B. W. Brewer, On primes of the form $u^{2}+5v^{2}$, Proc. Amer. Math. Soc. 17 (1966), 502–509. MR 188171, DOI 10.1090/S0002-9939-1966-0188171-6
Paromita Chowla, A new proof and generalizaion of some theorems of Brewer, Norske Vid. Selsk. Forh. (Trondheim) 41 (1968), 1–3. MR 252353
S. Chowla, A formula similar to Jacobsthal’s for the explicit value of $x$ in $p=x^2+y^2$ where $p$ is a prime of the form $4k+1$, Proc. Lahore Philos. Soc. 7 (1945), 2. MR 14120

The last entry in Gauss’s diary

35

10

Helmut Hasse, Über die Teilbarkeit durch $2^{3}$ der Klassenzahl imaginärquadratischer Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilern, J. Reine Angew. Math. 241 (1970), 1–6 (German). MR 258792, DOI 10.1515/crll.1970.241.1
L. E. Dickson, Cyclotomy, Higher Congruences, and Waring’s Problem, Amer. J. Math. 57 (1935), no. 2, 391–424. MR 1507083, DOI 10.2307/2371217
L. E. Dickson, Cyclotomy and trinomial congruences, Trans. Amer. Math. Soc. 37 (1935), no. 3, 363–380. MR 1501791, DOI 10.1090/S0002-9947-1935-1501791-3
L. E. Dickson, Cyclotomy when $e$ is composite, Trans. Amer. Math. Soc. 38 (1935), no. 2, 187–200. MR 1501808, DOI 10.1090/S0002-9947-1935-1501808-6

Zur Theorie der quadratischen Zerfallung der Primzahlen $8n + 3,7n + 2$ und

37

Über die Darstellung der Primzahlen der Form $4n + 1$ als Summe zweier Quadrate

132

Emma Lehmer, On the number of solutions of $u^k+D\equiv w^2(\mod p)$, Pacific J. Math. 5 (1955), 103–118. MR 67918
William Judson LeVeque, Topics in number theory. Vols. 1 and 2, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956. MR 0080682
Joseph B. Muskat, On Jacobi sums of certain composite orders, Trans. Amer. Math. Soc. 134 (1968), 483–502. MR 232752, DOI 10.1090/S0002-9947-1968-0232752-4

Mémoire sur les lois de réciprocité relatives aux résidues de puissances

31

S. F. Robinson, Theorems on Brewer sums, Pacific J. Math. 25 (1968), 587–596. MR 227121, DOI 10.2140/pjm.1968.25.587

Eine Beweis für die Zerlegbarkeit der Primzahlen von der Form $6n + 1$ in ein einfaches und ein dreifaches Quadrat

140

L. Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Math. Ann. 37 (1890), no. 3, 321–367 (German). MR 1510649, DOI 10.1007/BF01721360

Finite Fourier series and Jacobsthal sums

André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Publ. Inst. Math. Univ. Strasbourg, vol. 7, Hermann & Cie, Paris, 1948 (French). MR 0027151
Albert Leon Whiteman, Cyclotomy and Jacobsthal sums, Amer. J. Math. 74 (1952), 89–99. MR 45758, DOI 10.2307/2372071
A. L. Whiteman, The sixteenth power residue character of $2$, Canad. J. Math. 6 (1954), 364–373. MR 62761, DOI 10.4153/cjm-1954-035-8
Albert Leon Whiteman, The cyclotomic numbers of order ten, Proc. Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence, R.I., 1960, pp. 95–111. MR 0113851
A. L. Whiteman, The cyclotomic numbers of order twelve, Acta Arith. 6 (1960), 53–76. MR 118709, DOI 10.4064/aa-6-1-53-76
Albert Leon Whiteman, A theorem of Brewer on character sums, Duke Math. J. 30 (1963), 545–552. MR 154857
Albert Leon Whiteman, Theorems on Brewer and Jacobsthal sums. I, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 44–55. MR 0175861
Albert Leon Whiteman, Theorems on Brewer and Jacobsthal sums. II, Michigan Math. J. 12 (1965), 65–80. MR 217037