Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Thaine, F.

doi:10.1090/S0025-5718-99-01142-4

Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers
HTML articles powered by AMS MathViewer

by F. Thaine PDF

Math. Comp. 69 (2000), 1653-1666 Request permission

Abstract:

Given an odd prime $p$ we show a way to construct large families of polynomials $P_{q}(x)\in \mathbb {Q}[x]$, $q\in \mathcal {C}$, where $\mathcal {C}$ is a set of primes of the form $q\equiv 1$ mod $p$ and $P_{q}(x)$ is the irreducible polynomial of the Gaussian periods of degree $p$ in $\mathbb {Q}(\zeta _{q})$. Examples of these families when $p=7$ are worked in detail. We also show, given an integer $n\geq 2$ and a prime $q\equiv 1$ mod $2n$, how to represent by matrices the Gaussian periods $\eta _{0},\dots ,\eta _{n-1}$ of degree $n$ in $\mathbb {Q}(\zeta _{q})$, and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of $\mathbb {Q}(\eta _{0})$.

References

L.E. Dickson, Cyclotomy, higher congruences and Waring’s problem, Amer. J. Math. 57 (1935), 391–424.
Kenkichi Iwasawa, A note on Jacobi sums, Symposia Mathematica, Vol. XV (Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973) Academic Press, London, 1975, pp. 447–459. MR 0384747
Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
V. A. Lebesgue, Recherches sur les nombres, J. Math. Pures Appl. 2 (1837), 253–292.
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089
Thomas Storer, Cyclotomy and difference sets, Lectures in Advanced Mathematics, No. 2, Markham Publishing Co., Chicago, Ill., 1967. MR 0217033
H.W. Lloyd Tanner, On the binomial equation $x^{p}-1=0$: quinquisection, Proc. London Math. Soc. 18 (1886/87), 214–234.
F. Thaine, Properties that characterize Gaussian periods and cyclotomic numbers, Proc. Amer. Math. Soc. 124 (1996), no. 1, 35–45. MR 1301532, DOI 10.1090/S0002-9939-96-03108-5
F. Thaine, On the coefficients of Jacobi sums in prime cyclotomic fields, Transactions of the American Mathematical Society, to appear.
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7