On the order modulo p of an algebraic number (for p large enough)

Georges Gras (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

Let $K/\Q$ be Galois of degree $n$, of Galois group $G$, and let $\eta\in K^\times$ be given such that $\langle \eta \rangle_G \otimes \Q \simeq \Q[G]$. For $p$ unramified in $K/\Q$ and prime to $\eta$, denote by $n_p$ the residue degree of $p$, by $g_p$ the number of prime ideals ${\mathfrak p} \mid p$ then by $o_{\mathfrak p}(\eta)$ and $o_p(\eta)$ the orders of $\eta$ modulo ${\mathfrak p}$ and $p$, respectively. In a first part, using Frobenius automorphisms, we show that for all $p$ large enough, some explicit divisors of $p^{n_p}-1$ cannot realize $o_{\mathfrak p}(\eta)$ (Thms. 2.1, 4.1). In a second part, we obtain that for all $p$ large enough such that $n_p > 1$ we have ${\rm Prob}(o_p(\eta) < p) \leq \frac{1}{p^{g_p\,(n_p-1) - \varepsilon}}$, where $\varepsilon = O \big( \frac{1}{{\rm log}_2(p)}\big)$ (Thm. 6.1). Thus, under the heuristic of Borel--Cantelli this yields $o_p(\eta) > p$ for all $p$ large enough such that $g_p(n_p-1) \geq 2$, which covers the particular cases of cubic fields with $n_p=3$ and quartic fields with $n_p=2$, but not the quadratic fields with $n_p=2$; in this case, the natural conjecture is, on the contrary, that $o_p(\eta) < p$ for infinitely many inert $p$ (Conj. 8.1).

Mots clés: algebraic numbers; order modulo a prime; Frobenius automorphisms

Domaines: Théorie des nombres [math.NT]

Georges Gras. On the order modulo p of an algebraic number (for p large enough). 2015. ⟨hal-01225061v1⟩