A valuation criterion for normal bases in elementary abelian extensions

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Let p be a prime number and let K be a finite extension of the field Qₚ of p-adic numbers. Let N be a fully ramified, elementary abelian extension of K. Under a mild hypothesis on the extension N/K, we show that every element of N with valuation congruent mod [N:K] to the largest lower ramification number of N/K generates a normal basis for N over K.