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On rationality of Jacobi sums. (English) Zbl 0969.11029

From the introduction: Let \(p\) be an odd prime and \(q=p^f\), where \(f\) is a positive integer. Let \(\text{GF}(q)\) be the finite field of \(q\) elements. The character group of the multiplicative group \(\text{GF} (q)^\times\) is generated by the Teichmüller character \(\omega\), and is cyclic of order \(q-1\).
Let \(\eta\in \langle \omega\rangle\) be a nonprincipal character. For any character \(\chi\in\langle \omega\rangle\) different from the principal character \(\omega^0\) and from the character \(\eta\) we consider the Jacobi sum \[ J(\chi,\eta) =\sum_{x\in \text{GF}(q)- \{0,1\}} \chi(x) \eta(1-x). \] We consider the problem of obtaining precise conditions to ensure that \(J (\chi,\eta)\) belongs to the rational number field \(\mathbb{Q}\). This problem seems to be of interest in itself and has an application. Indeed, it is related to a question in algebraic combinatorics. The Jacobi sum \(J(\chi,\eta)\) with the quadratic character \(\eta= \omega^{q-1 \over 2}\) belongs to \(\mathbb{Q}\) if and only if the \(T\)-submodule of the Terwilliger algebra obtained from a cyclotomic scheme with class 2 is reducible.
In this paper we treat only the case where the character \(\eta\) is the quadratic character \(\omega^{q-1 \over 2}\). Namely we determine conditions on \(\chi\) and \(q\) ensuring that \(J(\chi,\eta)\) belongs to the rationals \(\mathbb{Q}\), in the case \(f=2\):
Suppose \(q=p^2\) and \(1\leq i\leq p^2-1\). Then \(J(\omega^{-i}, \omega^{p^2-1 \over 2})\) is rational if only if \(i=(p-1)k\) (\(k=1,2,\dots,p\)), or \(i={p+1\over 2}k\) (\(k=1,2, \dots,2(p-1)-1\)), or \(\omega^{-i}\) is of order 24 and \(p\equiv 17,19\pmod{24}\), or \(\omega^{-i} \) is of order 60 and \(p\equiv 41,49\pmod{60}\). We can discuss the problem in the general case by the same method.
S. Akiyama [Acta Arith. 75, 97-104 (1996; Zbl 0849.11094)] has independently obtained the same result with a completely different proof. The authors of the present paper had already announced their result in a symposium of RIMS at Kyoto University held in November 1994.

MSC:

11L10 Jacobsthal and Brewer sums; other complete character sums
11T24 Other character sums and Gauss sums

Citations:

Zbl 0849.11094
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