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On wild ramification in quaternion extensions. (English) Zbl 1123.11037

As the authors point out at the beginning, quaternion extensions are popular as test objects in Galois module structure and elsewhere (it suffices to mention early work of J. Martinet [Ann. Sci. Éc. Norm. Supér. (4) 4, 399–408 (1971; Zbl 0219.12012)] and A. Fröhlich [Invent. Math. 17, 143–166 (1972; Zbl 0261.12008)] on rings of integers in such extensions, and the “Quaternionic Exercises” of T. Chinburg et al. [Fields Inst. Commun. 16, 1–29 (1997; Zbl 0886.11063), Algebraic \(K\)-theory and its applications. Proceedings of the workshop and symposium, ICTP, Trieste, Italy, September 1–19, 1997. Singapore: World Scientific. 337–369 (1999; Zbl 0983.11068)].
In the present paper, the authors completely describe the possible ramification patterns which occur for totally (wildly) ramified quaternion extensions \(N\) of dyadic fields \(K\) which contain a fourth root of unity. Good use is made of a recent innovation due to the first author, the so-called refined ramification filtration. This is useful if the biquadratic extension \(M/K\) inside \(N\) is too homogeneous, that is, all three quadratic subfields have the same break number. The methods are very explicit, and rather technical. The extension \(N/M\) is obtained by adjoining a square root of an element of \(M\) which is subject to many constraints. Its “defect” (loosely speaking, its 2-adic distance to 1) determines the break number of \(N/M\); of course the possible values here depend on the break numbers (classical or refined) of \(M/K\). It is not feasible to precisely describe the results in a review. Suffice it to say that they are complete, well explained, and that they allow to describe with precision just when the Hasse-Arf theorem (integrality of break numbers in the upper numbering) fails for \(N/K\); as the authors explain, this is a rare event. It might be interesting to look at this phenomenon for Galois groups which are (with apologies for the expression) “even less commutative” than the quaternion group.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
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References:

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