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2nd level. Here the roots Ta1 a2 are not (algebraically) independent. For example, the following identity −a1 (Ta1 Ta2 ) p = (Ta1 +t0 −a2 )(Ta2 +t0 −a1 ) = Ta1 Ta2 +t0 −a2 Ta2 +t0 −(a1 +a2 ) Ta1 +t0 implies under the assumption (a1 + a2 , p) = 1 (and after a suitable choice of involved roots of Artin-Schreier equations) that Ta1 Ta2 = Ta1 a2 + Ta2 a1 + Ta1 +a2 . 8 Victor Abrashkin The presence of the term Ta1 +a2 creates a problem: τa1 +a2 should act nontrivially on either Ta1 a2 or Ta2 a1 but they both do not depend on the index a1 + a2 .

As Da1 . . Das ∈ AKsep 1 s< p ai ∈Z0 ( p) Define the diagonal map as the morphism of F p -algebras : A mod deg p −→ A ⊗ A mod deg p such that for any a ∈ Z0 ( p), Da → Da ⊗ 1 + 1 ⊗ Da . Then we have the following properties: — (E) ≡ E ⊗ E mod deg p; (F) ≡ F ⊗ Fmod deg p ; — σ (F) ≡ EF mod deg p; τa (F ) ≡ F exp(Da ) mod deg p . Now we can verify the existence of f ∈ LKsep such that F = exp( f ) modulo deg p, and recover the basic relations σ ( f ) = ( a t0−a Da ) ◦ f and τa ( f ) = f ◦ Da , a ∈ Z0 ( p), of our nilpotent Artin-Schreier theory.

6) can be deduced from the Brückner-Vostokov explicit reciprocity law. 4 Differentiation adτˆ0 ∈ Diff(L M ) It can be proved that there are only finitely many different ideals L(v) M mod J M . Therefore, we can fix sufficiently large natural number N1 (which depends only on N0 , e K and M), set Fγ0 := Fγ0,−N1 and use these elements to describe (v) the ramification filtration {L M modJ M }v≥0 . The elements Fγ0 , γ > 0, can be given modulo the ideal of third commutators C3 (L M k ) as follows : if γ = apl ∈ N, where a ∈ Z+ ( p) and l ∈ Z 0 , then Fγ0 = apl Dal¯ + η(n)a1 p s [Da1 s¯ , Da2 s¯−n¯ ]; s,n,a1 ,a2 Galois groups of local fields, Lie algebras and ramification 19 if γ ∈ / N, then F(γ ) = η(n)a1 ps [Da1 ,¯s , Da2 ,¯s −n¯ ].