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On the Computation of Hilbert Class Fields

1998
*
Journal of Number Theory
*

Let k be

doi:10.1006/jnth.1997.2208
fatcat:sdmbalcffngl3o47jufrchj7ci
*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. ...##
###
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]

2010
*
Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
*

This paper analyzes

doi:10.1137/1.9781611973075.40
dblp:conf/soda/EisentragerH10
fatcat:mzn3u2vqonfphcyav23jalbiuq
*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*. ...##
###
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

1999
*
Journal de Théorie des Nombres de Bordeaux
*

Finally,

doi:10.5802/jtnb.235
fatcat:kltfatwlurh6rdfdkinm34wpce
*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 ...##
###
Stark's Conjectures and Hilbert's Twelfth Problem

2000
*
Experimental Mathematics
*

As a direct

doi:10.1080/10586458.2000.10504650
fatcat:2odyunvo3bfpzgxhacw6lma5ia
*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. ...##
###
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

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. ...

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

2005
*
Lecture Notes in Computer Science
*

A crucial step

doi:10.1007/11496618_20
fatcat:7jit2yotgrdohcazqpnnpubdcy
*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 . ...##
###
Characteristic Numbers and Generalized Path Integrals
[article]

1994
*
arXiv
*
pre-print

Here we discuss characteristic

arXiv:dg-ga/9406002v1
fatcat:v4hva7afujcgppqrv3snhiz63a
*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 ...##
###
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]

2008
*
arXiv
*
pre-print

In this paper, we propose

arXiv:0804.1652v1
fatcat:zwhb2i7vxfbnlltg2lfva6qrgy
*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 ...##
###
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

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> ...

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