A valuation criterion for normal basis generators in local fields of characteristic p

Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with ${v_L(L^{\times})=\mathbb{Z} }Let K be a complete local field of characteristic p with perfect residue field, and let L/K be a finite, totally ramified, Galois p-extension with G = Gal(L/K). Let v _L be the normalized valuation with v_L(L^×)=\mathbbZ {v_L(L^{\times})=\mathbb{Z} }. Let p_L ? L{\pi_L\in L} be a prime element, and let p′ (x) be the derivative of the minimal polynomial for π _L over K. We show that any element r ? L{\rho\in L} with v_L(r) o -v_L(p￠(p_L))-1 mod L:K]{v_L(\rho)\equiv -v_L(p'(\pi_L))-1\bmodL:K]} generates a normal basis: KG]ρ = L. This criterion is tight: Given any integer i with i\not o -v_L(p￠(p_L))-1 mod L:K]{i\not\equiv -v_L(p'(\pi_L))-1\bmodL:K]}, there is a r_i ? L{\rho_i\in L} with v _L (ρ _i ) = i such that KG]r_i\subsetneq L{KG]\rho_i\subsetneq L}.