Canonical invariants for corresponding residue systems in P-adic fields

Abstract

Let F be a finite extension of Q p and let L F be a totally ramified, normal extension of degree p 2 n with normal subextensions K F and K′ F satisfying K ∩ K′ = F, KK′ = L, and [ K: F] = [ K′: F]. Let B be the maximal ideal of D L . Suppose t 1( K F ) = t 1( K′ F ) = 1 , where t 1 denotes the first breakpoint in the Hilbert ramification sequence for the extensions. We introduce canonical invariants of the extensions L/K/F and L/K′/F which determine M L ( K, K′), the maximal integer m so that D K + B m = D K′ + B m . In addition, we show that if π and π′ are prime elements of D K and D K′ , then υ L(π−π′)>ν L(π)⇔ν L(π−π′)=M L(K,K′). As a final consequence, we show that M L(K,K′)⩽ max{2p n−|G 2|,2p n−|G′ 2|}, where G 2 and G′ 2 are the 2nd ramification subgroups of G = Gal( K F ) and G′ = Gal( K′ F ) , sharpening a previous upper bound given for M L ( K, K′) under these hypotheses.