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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

Software:

GAP; SageMath
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References:

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