On Kummer extensions of the power series field
نویسنده
چکیده
In this paper we study the Kummer extensions of a power series field K = k((X1, ...,Xr)), where k is an algebraically closed field of arbitrary characteristic. 1 Terminology and notation Let k be an algebraically closed field, X1, ...,Xr indeterminates formally independent over k, and let K and Lm be the fields K = k ((X1, ...,Xr)) , Lm = k (( X 1/m 1 , ...,X 1/m r )) , where m is a non negative integer, not divisible by the characteristic of k. The extension K ⊂ Lm is trivially normal, finite and separable, its Galois group being G ≃ (Cm), where Cm stands for the cyclic group of m elements. The elements of G will be noted (a1, ..., ar) : Lm −→ Lm, 0 ≤ ai < m Xl 7−→ ωlXl where ω ∈ k is an m–th primitive root of the unity. Let R and Sm be the rings R = k [[X1, ...,Xr ]] , Sm = k [[ X 1/m 1 , ...,X 1/m r ]] . ∗Supported by JdA (FQM 218) and MCyT (BFM2001-3207 and FEDER).
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