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Degree 12 2-adic fields with automorphism group of order 4. (English) Zbl 1348.11090

Let \({\mathbb Q}_2\) denote the field of 2-adic numbers. It is well known that for every \(n\geq1\) there are only finitely many isomorphism classes of extensions \(K/{\mathbb Q}_2\) such that \([K:{\mathbb Q}_2]=n\). The extensions of \({\mathbb Q}_2\) of degree \(n\leq11\) have been completely classified. This paper describes a method for determining the extensions \(K/{\mathbb Q}_2\) of degree 12 such that Aut\((K/{\mathbb Q}_2)\) has order 4. The authors report that there are 513 such fields. They also state that they have determined irreducible polynomials of degree 12 over \({\mathbb Q}_2\) which correspond to each of these fields, and they give numerous examples of these polynomials. Their methods allow them to compute the ramification index and discriminant for each extension \(K/{\mathbb Q}_2\), as well as the Galois group Gal\((\widetilde{K}/{\mathbb Q}_2)\) of the normal closure \(\widetilde{K}/{\mathbb Q}_2\) of \(K/{\mathbb Q}_2\).

MSC:

11S15 Ramification and extension theory
11S05 Polynomials
11S20 Galois theory

Software:

GAP
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References:

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