On the coefficients of the minimal polynomials of Gaussian periods

Abstract:

Let l be a prime number and m a divisor of $l - 1$. Then the Gauss period $\omega = \zeta + {\zeta ^\lambda } + {\zeta ^{{\lambda ^2}}} + \cdots + {\zeta ^{{\lambda ^{m - 1}}}}$ where $\zeta = {e^{2\pi i/l}}$ and $\lambda$ is a primitive mth root of unity modulo l, generates a subfield K of $\mathbb {Q}(\zeta )$ of degree $(l - 1)/m$. In this paper we study the reciprocal minimal polynomial ${F_{l,m}}(X) = {N_{K/\mathbb {Q}}}(1 - \omega X)$ of $\omega$ over $\mathbb {Q}$. It will be shown that for fixed m and every N we have ${F_{l,m}}(X) \equiv {({B_m}{(X)^l}/(1 = mX))^{1/m}}\;\pmod {{X^N}}$ for all but finitely many "exceptional primes" l (depending on m and N), where ${B_m}(X) \in \mathbb {Z}[[X]]$ is a power series depending only on m . A method of computation of this set of exceptional primes is presented. The generalization of the results to the case of composite l is also discussed.