## 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

### Citations:

Zbl 0332.10003; Zbl 0539.10006
Full Text:

### References:

  Hasse, H., Vorlesungen uber Zahlentheorie, Berlin1950. · Zbl 0038.17703  Lehmer, E., The quintic character of 2 and 3, Duke math. J.18 (1951), 11-18. · Zbl 0045.02002  Leonard, P.A., Mortimer, B.C. and Williams, K.S., The eleventh power character of 2, Crelle286/287 (1976), 213-222. · Zbl 0332.10003  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  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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