****************************************************************************** GTM193: Advanced Topics in Computational Number Theory ****************************************************************************** Chapter 1 ------------------------------------------------------------------------------ Extended Euclid in Dedekind idealaddtoone Approximation Theorem idealappr, idealchinese Random Element in Ideal idealtwoelt Coprime Ideal Class idealcoprime Two-Element Representation idealtwoelt ad-bc Algorithm in nfhnf Hermite Normal Form in Dedekind nfhnf HNF Reduction Modulo Ideal nfreducemodideal LLL Reduction Modulo Ideal lllreducemodmatrix Intersection of Z-Modules idealintersect Reduction mod p of PseudoBasis --- Hermite Normal Form mod D in Dedekind nfhnfmod, nfdetint Smith Normal Form in Dedekind nfsnf ------------------------------------------------------------------------------ Chapter 2 ------------------------------------------------------------------------------ Compositum Using t1+kt2 polcompositum, nfcompositum Compositum Using t1t2+kt2 --- Relative to Absolute rnfequation Reversion of Algebraic Number modreverse Compositum with Normal Extension --- Small HNF PseudoMatrix of Ideal --- PseudoTwoElement Representation rnfidealtwoelt Valuation at a Relative Prime rnfidealfactor Relative Prime Ideal Inversion --- Relative Ideal Inversion --- Ideal Factorization idealfactor Ideal List ideallist Squarefree Ideal List --- Conductor at l Ideal List --- Relative Dedekind Criterion rnfdedekind Relative Round 2 rnfpseudobasis Relative Discriminant rnfdisc Relative Basis, if Exists rnfbasis Steinitz Class rnfsteinitz Is PseudoBasis Free rnfisfree Relative Polynomial Reduction rnfpolred, rnfpolredabs Relative Prime Decomposition rnfidealprimedec Absolute to Relative Element rnfeltabstorel Relative to Absolute Element rnfeltreltoabs Absolute to Relative Ideal rnfidealabstorel Relative to Absolute Ideal rnfidealreltoabs Relative Norm of Ideal rnfidealnormrel Absolue Norm of a Relative Ideal rnfidealnormabs Ideal Up rnfidealup Ideal Down rnfidealdown ------------------------------------------------------------------------------ Chapter 3 (no algorithm) ------------------------------------------------------------------------------ Chapter 4 ------------------------------------------------------------------------------ SNF for Groups smithrel Quotient of Groups InitQuotient [stark.c] Group Extension --- Right 4-Term Exact Sequence --- Image of a Group --- Inverse Image of a Group --- Kernel of a Group Homomorphism ComputeKernel [stark.c] Cokernel of a Group Homomorphism --- Left 4-Term Exact Sequence --- Intersection and Sum of Subgroups --- Intersection in a Subgroup --- p-Sylow Subgroup --- Enumeration of Subgroups subgrouplist, forsubgroup Subgroups of Index l subgrouplist, forsubgroup Linear System in Integers mathnf + hnf_gauss Linear System of Congruences matsolvemod Mixed Linear System mathnf + hnf_gauss + matkerint Chinese for Ideals idealchinese OneElement Representation in (Z_K/m)^* set_sign_mod_idele [base3.c] Computation of (Z_K/m)^* idealstar Coprime Representative idealcoprime Discrete Logarithm in (Z_K/m)^* ideallog Ray Class Group bnrclass, bnrclassno, bnrinit Principal Ideal in Ray Class Group bnrisprincipal Reduction of an Ideal [ useless, see famat_to_nf_modideal_coprime ] Reduction of Ray Ideal Class [ useless, see famat_to_nf_moddivisor ] Computation of CP_n --- Conductor of a Congruence Subgroup bnrconductor Is Modulus Conductor bnrisconductor Norm Group of Abelian Extension rnfnormgroup Conductor of Abelian Extension rnfconductor Norm Group of Abelian Extension rnfnormgroup Is Extension Abelian rnfisabelian Conductor of Character bnrconductorofchar ------------------------------------------------------------------------------ Chapter 5 ------------------------------------------------------------------------------ Splitting Class Field Extensions --- Decomposition of an Ideal in Cl(K)/Cl(K)^l --- Kummer Extension when Zeta in K, Hecke rnfkummer (rnfkummersimple) Discrete Log in Unit Group bnfisunit Discrete Log in Selmer Group --- Kummer Extension when Zeta not in K, Hecke rnfkummer Action of Artin on Kummer --- Kummer Extension when Zeta in K, Artin bnrclassfield Kummer Extension when Zeta not in K, Artin bnrclassfield ------------------------------------------------------------------------------ Chapter 6 ------------------------------------------------------------------------------ Compute Root Number W(chi) bnrrootnumber Compute Values of L(1,chi) or Order bnrL1 Hilbert Class Field of Real Quadratic Field quadhilbert Ray Class Field of Real Quadratic Field quadray Real Ray Class Field bnrstark P in Z[X] for Hilbert Class Fields makescind (internal) Eta function eta Hilbert Class Field of Imaginary Quad. Field quadhilbert List of Reduced Forms in quadhilbert Quasi-Periods elleta, ell.eta Weierstrass Zeta Function ellzeta Weierstrass Sigma function ellsigma Ray Class Field of Imaginary Quadratic Field quadray ------------------------------------------------------------------------------ Chapter 7 ------------------------------------------------------------------------------ Cl_i(L/K), Cl_i(K) --- U(L)/i(U(K)) --- U(L)/(mu(L) i(U(K))) --- U_N(L/K) --- Relative Ideal Reduction --- S-Class Group bnfsunit S-Unit Group bnfsunit Discrete Log in S-Unit Group bnfissunit Solve Relative Norm Equation rnfisnorm Solve Relative Integral Norm Equation bnfisintnorm [ absolute ] ------------------------------------------------------------------------------ Chapter 8 ------------------------------------------------------------------------------ Cubic Form Test --- Real Cubic Field Table --- Complex Cubic Field Table --- ------------------------------------------------------------------------------ Chapter 9 ------------------------------------------------------------------------------ Quadratic Extensions Squarefree Ideals --- Quadratic Extensions by Class Field Theory --- Relative Cyclic Cubic Extensions --- Relative Non-Cyclic Cubic Extensions --- Quartic Fields Using Geometry of Numbers --- ------------------------------------------------------------------------------ Chapter 10 ------------------------------------------------------------------------------ Solving l-th Power Congruences sqrtn Solving l-th Power Congruences k<=e --- Dirichlet Series with Functional Equation elllseries