منابع مشابه
Number Fields Unramified Away from 2
Consider the set of number fields unramified away from 2, i.e., unramified outside {2,∞}. We show that there do not exist any such fields of degrees 9 through 15. As a consequence, the following simple groups are ruled out for being the Galois group of an extension which is unramified away from 2: Mathieu groups M11 and M12, PSL(3, 3), and alternating groups Aj for 8 < j < 16 (values j ≤ 8 were...
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The first mathematician who studied quaternion extensions (H8-extensions for short) was Dedekind [6]; he gave Q( √ (2 + √ 2)(3 + √ 6) ) as an example. The question whether given quadratic or biquadratic number fields can be embedded in a quaternion extension was extensively studied by Rosenblüth [32], Reichardt [31], Witt [36], and Damey and Martinet [5]; see Ledet [19] and the surveys [15] and...
متن کاملRemark on infinite unramified extensions of number fields with class number one
We modify an idea of Maire to construct biquadratic number fields with small root discriminants, class number one, and having an infinite, necessarily non-solvable, strictly unramified Galois extension. Let k be an algebraic number field with class number one. Then k has no Abelian (and hence no solvable) non-trivial unramified Galois extension. It is somewhat surprising that k may nevertheless...
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Let p be prime and Zpn a degree n unramified extension of the ring of p-adic integers Zp. In this paper we give an overview of some very fast deterministic algorithms for common operations in Zpn modulo p . Combining existing methods with recent work of Kedlaya and Umans about modular composition of polynomials, we achieve quasi-linear time algorithms in the parameters n and N , and quasi-linea...
متن کاملUnramified Alternating Extensions of Quadratic Fields
We exhibit, for each n ≥ 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group An. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n = 5. It is a folk conjecture that for n ≥ 5, all but finitel...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2010
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2010.02.005