Explicit integral Galois module structure of weakly ramified extensions of local fields

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Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G₁ and G₂ be the first and second ramification groups. Thus L/K is tamely ramified when G₁ is trivial and we say that L/K is weakly ramified when G₂ is trivial. Let Oₗ be the valuation ring of L and let Pₗ be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G₁| then Pₗⁿ is free over the group ring Oₖ[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every free generator of Pₗ over Oₖ[G] is also a free generator of Oₗ over its associated order in the group algebra K[G]. Along the way, we prove a `splitting lemma' for local fields, which may be of independent interest.