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Let K/Q be a Galois extension of degree n, of Galois group G, and let$\eta\in K^\times$. For all large enough prime p, we define, by use of theFrobenius theorem on group determinants, the family $(\Delta_p^\theta(\eta) \in\F_p)_\theta$ of local $\theta$-regulators of $\eta$, indexed by theQp-irreducible characters $\theta$ of G. At each $\Delta_p^\theta (\eta)$ isassociated a linear representation $L^\theta \simeq \delta V_\theta$, $0 \leq\delta \leq \varphi(1)$, which characterizes some properties of$\Delta_p^\theta (\eta)$, including its nullity equivalent to $\delta \geq 1$(Th. 3.11). When $\eta \in \Q^\times$ and $\theta = 1$, $\Delta_p^1 (\eta)$ isthe p-Fermat quotient of $\eta$. When $\eta$ is a "Minkowski unit", each$\Delta_p^\theta (\eta)$, $\theta \ne 1$, gives the residue modulo p of the$\theta$-component of $p^{1-n} Reg_p (K)$, where Reg(K) is the classical p-adicregulator of K. We suggest that the "probability" of ($\Delta_p^\theta(\eta) =0$ and $L^\theta \simeq \delta V_\theta$) is $\frac{O(1)}{p^{f \delta^2}}$,where f is a suitable residue degree of p. We conjecture that $p^{1-n}Reg_p(K)$, which measures the order of the p-torsion group in Abelianp-ramification over K, is for p large enough a p-adic unit except perhaps for aset of prime numbers of zero density. For these cases said "of minimalp-divisibility" (Def. 3.17), it remains possible, $\eta$ being then a "partiallocal pth power" at p, to propose, in connection with the ABC conjecture, astronger conjecture leading to the same conclusion for all large enough p(Section 7). Some other conjectural aspects on the Fermat quotient arediscussed. We precise and verify these properties through numerical studies onvarious fields and publish the corresponding "PARI" programs.

Notion de $\theta$-r\'egulateurs d'un nombre alg\'ebrique. Conjectures p-adiques