Classifying extensions of the field of formal Laurent series over $$\mathbb{F}_p$$.(English)Zbl 1320.11110

Let $$K$$ be a local field and $$\overline K$$ an algebraic closure. It is well known that, for each $$n \in \mathbb N$$ not divisible by char$$(K)$$, the set of degree $$n$$ extensions of $$K$$ in $$\overline K$$ is finite, and a formula for their number has been deduced in characteristic zero by M. Krasner [Colloq. Int. CNRS 143, 143–169 (1966; Zbl 0143.06403)]. The paper under review continues the study of these extensions along the lines drawn by J. W. Jones and D. P. Roberts, for degree $$n$$ extensions of $$\mathbb Q _ p$$, for small values of $$p$$ and $$n$$ (see [J. Symb. Comput. 41, No. 1, 80–97 (2006; Zbl 1140.11350)]).
The authors consider the case where char$$(K) = p > 0$$, which allows them to identify $$K$$ with a finite extension of the formal Laurent power series field $$\mathbb F _ p((T))$$ in a variable $$T$$ over the field $$\mathbb F _ p$$ with $$p$$ elements. They present a simple method of describing the extensions of $$K$$ in $$K _ {\text{sep}}$$ of degree $$n$$. The method is based on well-known properties of the standard discrete valuation $$v$$ of $$K$$. Firstly, on the unique, up-to an equivalence, extendability of $$v$$ to a valuation $$v _ L$$ on each finite extension $$L$$ of $$K$$ in $$\overline K$$. Secondly, on the fundamental equality $$[L: K] = [\widehat L: \widehat K]e(L/K)$$, where $$\widehat K$$ and $$\widehat L$$ are the residue fields of $$(K, v)$$ and $$(L, v _ L)$$, respectively, and $$e(L/K)$$ is the ramification index of $$L/K$$, i.e., the index of the value group $$v(K)$$ in the value group $$v _ L(L)$$. Thirdly, on the following facts: $$L$$ includes as a subfield the unique unramified extension $$Y = Y(\widehat L)$$ of $$K$$ in $$\overline K$$ of degree $$[\widehat L: \widehat K]$$; $$L$$ is generated over $$Y$$ by an $$e$$-th root of a uniformizer of $$(Y, v _ Y)$$, in case $$[L: Y] = e = e(L/K)$$ and $$p$$ does not divide $$e$$; when $$L/K$$ is a Galois extension, the Galois group $$\mathcal{G}(L/K)$$ is solvable. Under the hypothesis that $$p \nmid e(L/K)$$, the suggested method allows the authors to determine the minimal polynomial over $$K$$ of some primitive element of $$L/K$$, and in the case where $$L/K$$ is Galois, to compute $$\mathcal{G}(L ^ {\prime }/K)$$. This is explicitly illustrated in the case of $$p = 3$$, for all degree $$10$$ extensions of $$K$$. The paper also contains a proof of a well-known fact (see, for example, A. Weil’s footnote on page 280 of the Russian translation [Moskau: “Mir” (1972; Zbl 0238.12001)] of his book [Basic number theory. Berlin-Heidelberg-New York: Springer Verlag (1967; Zbl 0176.33601)]).

MSC:

 11S05 Polynomials 11S15 Ramification and extension theory 11Y40 Algebraic number theory computations

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References:

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