Jacobi sums and new families of irreducible polynomials of Gaussian periods (Q2723533)

Let \(q\) be a prime \(\equiv 1 \bmod {2m}\), where \(m>2\), and let \(s\) be a primitive root mod \(q\). The author studies the Jacobi sums NEWLINE\[NEWLINE J_{a,b}= -\sum_{k=2}^{q-1}\zeta_m^{a{\text{ ind}}_s(k)+b{\text{ ind}} _s(1-k)}\quad(0\leq a,\;b\leq m-1), NEWLINE\]NEWLINE where \(\zeta_m\) is a primitive \(m\)th root of 1 and \(\operatorname {ind}_s(k)\) is the least nonnegative index of \(k\mod q\) for \(s\). The emphasis is on the computation of \(J_{a,b}\). The results are used to construct families of irreducible polynomials \(P_q(x)\) of Gaussian periods of degree \(m\) in the \(q\)th cyclotomic field, where \(q\) runs through a suitable set of primes. Examples of such families are exhibited for small \(m\). The author also provides MAPLE programs for calculating Jacobi sums and polynomials \(P_q(x)\). NEWLINENEWLINENEWLINEMany of the results of this article extend and complete the author's previous work [Trans. Am. Math. Soc. 351, 4769-4790 (1999; Zbl 0944.11036), Math. Comput. 69, 1653-1666 (2000; Zbl 0989.11055)]. The problem of calculating irreducible polynomials of Gaussian periods is related to an article of \textit{R. Schoof} and \textit{L. C. Washington} [Math. Comput. 50, 543-556 (1988; Zbl 0649.12007)] providing real cyclotomic fields with large class numbers.