Extensions of local fields and elementary symmetric polynomials

2016/08/26 by Keating, Kevin
#11S15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1608.07350

Abstract

Let K be a local field whose residue field has characteristic p and let L/K be a finite separable totally ramified extension of degree n=up^ν. Let σ₁,…,σ_n denote the K-embeddings of L into a separable closure K^sep of K. For 1≤ h≤ n let e_h(X₁,…,X_n) denote the hth elementary symmetric polynomial in n variables, and for α∈ L set E_h(α) =e_h(σ₁(α),…,σ_n(α)). Set j=min\v_p(h),ν\. We show that for r∈ℤ we have E_h(M_L^r)⊂ M_K^{\lceil(i_j+hr)/n\rceil}, where i_j is the jth index of inseparability of L/K. In certain cases we also show that E_h(M_L^r) is not contained in any higher power of M_K.

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