The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields
نویسندگان
چکیده
منابع مشابه
Class numbers of ray class fields of imaginary quadratic fields
Let K be an imaginary quadratic field with class number one and let p ⊂ OK be a degree one prime ideal of norm p not dividing 6dK . In this paper we generalize an algorithm of Schoof to compute the class numbers of ray class fields Kp heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura’s reciprocity law. We have...
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Motivated by a constructive realization of generalized dihedral groups as Galois groups over Q and by Atkin’s primality test, we present an explicit construction of the Hilbert class fields (ring class fields) of imaginary quadratic fields (orders). This is done by first evaluating the singular moduli of level one for an imaginary quadratic order, and then constructing the “genuine” (i.e., leve...
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The implementation of the Atkin-Goldwasser-Kilian primality testing algorithm requires the construction of the Hilbert class eld of an imaginary quadratic eld. We describe the computation of a de ning equation for this eld in terms of Weber's class invariants. The polynomial we obtain, noted W(X), has a solvable Galois group. When this group is dihedral, we show how to express the roots of this...
متن کاملNormal Bases of Ray Class Fields over Imaginary Quadratic Fields
We first develop a criterion to determine normal bases (Theorem 2.4), and by making use of necessary lemmas which were refined from [3] we further prove that singular values of certain Siegel functions form normal bases of ray class fields over all imaginary quadratic fields other than Q( √−1) and Q( √−3) (Theorem 4.5 and Remark 4.6). This result would be an answer for the Lang-Schertz conjectu...
متن کاملClass numbers of imaginary quadratic fields
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N . The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N . Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work ...
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2015
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042115500852