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On the Computation of Hilbert Class Fields

M. Daberkow, M. Pohst
1998 Journal of Number Theory  
Let k be an algebraic number field. We describe a procedure for computing the Hilbert class field 1(k) of k, i.e., the maximal abelian extension unramified at all places.  ...  In the sequel we consider an algebraic number field k and compute the maximal abelian extension 1(k) of k which is unramified at all places. This field 1(k) is called the Hilbert class field of k.  ...  ACKNOWLEDGMENT The authors thank H. Koch for various hints and comments which helped to improve this paper considerably.  ... 
doi:10.1006/jnth.1997.2208 fatcat:sdmbalcffngl3o47jufrchj7ci

Page 7668 of Mathematical Reviews Vol. , Issue 2000k [page]

2000 Mathematical Reviews  
In summary, the main topics of Advanced topics are relative extensions, class field theory, and number field table construction.  ...  Though both of these methods impose restrictions on the base field, they are much more efficient when they are applicable; in particular, they are efficient for the construction of Hilbert class fields  ... 

Page 5505 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
The first constructions considered are the ones from Hilbert class fields.  ...  The ray class field and Hilbert class field are obtained as applications of some results from class field theory such as the Artin reciprocity theorem and the conductor theorem.  ... 

Algorithms for ray class groups and Hilbert class fields [chapter]

Kirsten Eisenträger, Sean Hallgren
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
This paper analyzes the complexity of problems from class field theory. Class field theory can be used to show the existence of infinite families of number fields with constant root discriminant.  ...  We show that computing the ray class group and computing certain subfields of Hilbert class fields efficiently reduce to known computationally difficult problems.  ...  To show the existence of infinite families of number fields with constant root discriminant as described above one constructs an infinite tower of Hilbert class fields.  ... 
doi:10.1137/1.9781611973075.40 dblp:conf/soda/EisentragerH10 fatcat:mzn3u2vqonfphcyav23jalbiuq

Page 825 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
CM-fields with class number one which are Hilbert class fields of quadratic fields.  ...  In this note, the author gives without proof a list of quantitative results relative to certain Hilbert class fields of quadratic fields.  ... 

A survey of computational class field theory

Henri Cohen
1999 Journal de Théorie des Nombres de Bordeaux  
Finally, the C++ software library LiDIA from Darmstadt under the supervision of J . Buchmann is also a remarkable package, and will soon contain functions for class field theory.  ...  The KANT/KASH package from Berlin, under the supervision of M. Pohst.  ...  Apart from the intrinsic interest of computing ray class fields, the main application of the above algorithms is the construction of new number fields, in particular number fields of small discriminant  ... 
doi:10.5802/jtnb.235 fatcat:kltfatwlurh6rdfdkinm34wpce

Stark's Conjectures and Hilbert's Twelfth Problem

Xavier-François Roblot
2000 Experimental Mathematics  
As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples.  ...  extension of K.  ...  The eld we wish to construct is L = H K , the Hilbert Class Field of K, i.e. the maximal Abelian extension of K unrami ed everywhere, which is a cyclic extension of degree 4 of K.  ... 
doi:10.1080/10586458.2000.10504650 fatcat:2odyunvo3bfpzgxhacw6lma5ia

Page 687 of Mathematical Reviews Vol. , Issue 89B [page]

1989 Mathematical Reviews  
The author also in- troduces the conception of totally imaginary extension, obtains a relative class number formula for such extensions and an analogue of Kummer’s theorem 2| hj > 2|h>.  ...  First, for each natural number d > 0, he considers L, the ray class field modulo d, and defines a Stickelberger ideal S,,, which plays a role with respect to the ray class group Cz analogous to that of  ... 

p-Capitulation over Number Fields with p-Class Rank Two

Daniel C. Mayer
2016 Journal of Applied Mathematics and Physics  
An implementation of the complete algorithm in the computational algebra system Magma is employed for calculating the Artin pattern , K K K = τ AP( ) ( ( ) ( ))  of all 34631 real quadratic fields K d  ...  The results admit extensive statistics of the second 3-class groups K K = 2 3  ...  Acknowledgements The author gratefully acknowledges that his research is supported by the Austrian Science Fund (FWF): P 26008-N25.  ... 
doi:10.4236/jamp.2016.47135 fatcat:en5hr7sze5appm4dhivcnxvbzu

Generating Prime Order Elliptic Curves: Difficulties and Efficiency Considerations [chapter]

Elisavet Konstantinou, Aristides Kontogeorgis, Yannis C. Stamatiou, Christos Zaroliagis
2005 Lecture Notes in Computer Science  
A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones, uniquely determined by the CM discriminant  ...  In attempting to construct prime order ECs using Weber polynomials two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts  ...  This root can be used to construct the parameters of an EC with order m over the field F p .  ... 
doi:10.1007/11496618_20 fatcat:7jit2yotgrdohcazqpnnpubdcy

Characteristic Numbers and Generalized Path Integrals [article]

Daniel S. Freed
1994 arXiv   pre-print
Here we discuss characteristic numbers, particularly the Euler number of a complex line bundle over an oriented surface.  ...  In the second part of the paper we show how path integrals give rise to invariants which obey gluing laws.  ...  However, if we start with a 4 dimensional characteristic class, then the gerbe attached to a surface leads to a central extension of the diffeomorphism group of the surface which arises in quantum Chern-Simons  ... 
arXiv:dg-ga/9406002v1 fatcat:v4hva7afujcgppqrv3snhiz63a

Page 2667 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
More precisely, there is a first part where the Weil conjectures are used to compute Betti numbers by counting the points of the Hilbert schemes over finite fields.  ...  The last chapter deals with the description of the Chow ring of the relative Hilbert scheme of three points of a P? bundle.  ... 

Introducing Ramanujan's Class Polynomials in the Generation of Prime Order Elliptic Curves [article]

Elisavet Konstantinou, Aristides Kontogeorgis
2008 arXiv   pre-print
In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method.  ...  We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, M_D,l(x) and M_D,p_1,p_2(x) polynomials and show that they clearly outweigh all of them  ...  H(P j ) H(P f ) = deg f Φ(f, j) deg j Φ(f, j) = r(f ). (12) If f (τ ) does not generate the Hilbert class field but an algebraic extension of it with extension degree m then lim h(j(τ ))→∞ H(P j ) H(P  ... 
arXiv:0804.1652v1 fatcat:zwhb2i7vxfbnlltg2lfva6qrgy

Page 6865 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
Let & be a real quadratic number field with class number 2 and 2-class group in the strict sense of type (2,2).  ...  Let B be a quaternion algebra over a number field k, and let Q be an order in a quadratic extension L of k.  ... 

Some remarks on the construction of class polynomials

Elisavet Konstantinou, Aristides Kontogeorgis
2011 Advances in Mathematics of Communications  
Class invariants are singular values of modular functions which generate the class fields of imaginary quadratic number fields.  ...  Among all known class polynomials, Weber polynomials constructed with discriminants −D ≡ 1 (mod 8) have the smallest height and require the least precision for their construction.  ...  Acknowledgements We would like to thank the referees very much for their valuable comments and suggestions. <ekonstantinou@aegean.gr; kontogar@math.uoa.gr>  ... 
doi:10.3934/amc.2011.5.109 fatcat:ov6hdrwbsnamvcx5xav72beb2q
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