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Note on the Jacobi sum \(J(\chi,\chi)\). (English) Zbl 0853.11068

Let \(\zeta_\ell= \exp (2\pi i/\ell)\). Let \(p\) be a prime with \(p\equiv 1\pmod \ell\). Let \(g\) be a primitive root\(\pmod p\). Let \(\chi\) be the Dirichlet character \(\pmod p\) given by \(\chi(x)= \zeta_\ell^{\text{ind}_g(x)}\), \(x\not\equiv 0\pmod p\). Let \(J(\chi,\chi)\) be the Jacobi sum \(\sum^{p-1}_{x=0} \chi(x) \chi(1-x)\). The author addresses the question: When is \(J(\chi,\chi)\) uniquely determined up to units and conjugates by the solution of the equation \[ X\overline{X}= p,\quad X\in \mathbb{Z}[ \zeta_\ell], \quad x\equiv 1\pmod 2 ? \] He gives a complete solution in the cases \(\ell= 11\) and 19. On the basis of this result he gives a necessary and sufficient condition for 2 to be a 11th power \(\pmod p\), resp. a 19th power \(\pmod p\), when \(p\) is not representable by \(x^2+ 11y^2\), resp. \(x^2+ 19y^2\). These results complement earlier work of P. A. Leonard, B. C. Mortimer and K. S. Williams [J. Reine Angew. Math. 286/287, 213-222 (1976; Zbl 0332.10003)] and of J. C. Parnami, M. K. Agrawal, S. Pall and A. R. Rajwade [Acta Arith. 43, 361-365 (1984; Zbl 0539.10006)].

MSC:

11L10 Jacobsthal and Brewer sums; other complete character sums
11R18 Cyclotomic extensions
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References:

[1] Hasse, H., Vorlesungen uber Zahlentheorie, Berlin1950. · Zbl 0038.17703
[2] Lehmer, E., The quintic character of 2 and 3, Duke math. J.18 (1951), 11-18. · Zbl 0045.02002
[3] Leonard, P.A., Mortimer, B.C. and Williams, K.S., The eleventh power character of 2, Crelle286/287 (1976), 213-222. · Zbl 0332.10003
[4] Leonard, P.A. and Williams, K.S., The septic character of 2,3,5 and 7, Pacific J. Math.52 (1974), 143-147. · Zbl 0265.10004
[5] Parnami, J.C., Agrawal, M.K. and Rajwade, A.R., Criterion for 2 to be l-th power, Acta Arithmetica43 (1984), 361-364. · Zbl 0539.10006
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