1% Copyright (c) 2007-2016 Karim Belabas. 2% Permission is granted to copy, distribute and/or modify this document 3% under the terms of the GNU General Public License 4 5% Reference Card for PARI-GP, Algebraic Number Theory. 6% Author: 7% Karim Belabas 8% Universite de Bordeaux, 351 avenue de la Liberation, F-33405 Talence 9% email: Karim.Belabas@math.u-bordeaux.fr 10% 11% See refcard.tex for acknowledgements and thanks. 12\def\TITLE{Algebraic Number Theory} 13\input refmacro.tex 14\def\p{\goth{p}} 15 16\section{Binary Quadratic Forms} 17% 18\li{create $ax^2+bxy+cy^2$ (distance $d$) }{Qfb$(a,b,c,\{d\})$} 19\li{reduce $x$ ($s =\sqrt{D}$, $l=\floor{s}$)} 20 {qfbred$(x,\{\fl\},\{D\},\{l\},\{s\})$} 21\li{return $[y,g]$, $g\in \text{SL}_2(\ZZ)$, $y = g\cdot x$ reduced} 22 {qfbredsl2$(x)$} 23\li{composition of forms}{$x$*$y$ {\rm or }qfbnucomp$(x,y,l)$} 24\li{$n$-th power of form}{$x$\pow$n$ {\rm or }qfbnupow$(x,n)$} 25\li{composition without reduction}{qfbcompraw$(x,y)$} 26\li{$n$-th power without reduction}{qfbpowraw$(x,n)$} 27\li{prime form of disc. $x$ above prime $p$}{qfbprimeform$(x,p)$} 28\li{class number of disc. $x$}{qfbclassno$(x)$} 29\li{Hurwitz class number of disc. $x$}{qfbhclassno$(x)$} 30\li{solve $Q(x,y) = n$ in integers}{qfbsolve$(Q,n)$} 31 32\section{Quadratic Fields} 33% 34\li{quadratic number $\omega=\sqrt x$ or $(1+\sqrt x)/2$}{quadgen$(x)$} 35\li{minimal polynomial of $\omega$}{quadpoly$(x)$} 36\li{discriminant of $\QQ(\sqrt{x})$}{quaddisc$(x)$} 37\li{regulator of real quadratic field}{quadregulator$(x)$} 38\li{fundamental unit in real $\QQ(\sqrt{D})$}{quadunit($D$,\{'w\})} 39\li{class group of $\QQ(\sqrt{D})$}{quadclassunit$(D,\{\fl\},\{t\})$} 40\li{Hilbert class field of $\QQ(\sqrt{D})$}{quadhilbert$(D,\{\fl\})$} 41\li{\dots using specific class invariant ($D<0$)}{polclass$(D,\{\var{inv}\})$} 42\li{ray class field modulo $f$ of $\QQ(\sqrt{D})$}{quadray$(D,f,\{\fl\})$} 43\bigskip 44 45\section{General Number Fields: Initializations} 46The number field $K = \QQ[X]/(f)$ is given by irreducible $f\in\QQ[X]$. 47We denote $\theta = \bar{X}$ the canonical root of $f$ in $K$. 48A \var{nf} structure contains a maximal order and allows operations on 49elements and ideals. A \var{bnf} adds class group and units. A \var{bnr} is 50attached to ray class groups and class field theory. A \var{rnf} is attached 51to relative extensions $L/K$.\hfill\break 52% 53\li{init number field structure \var{nf}}{nfinit$(f,\{\fl\})$} 54\beginindentedkeys 55 \li{known integer basis $B$}{nfinit$([f,B])$} 56 \li{order maximal at $\var{vp}=[p_1,\dots,p_k]$}{nfinit$([f,\var{vp}])$} 57 \li{order maximal at all $p \leq P$}{nfinit$([f,P])$} 58 \li{certify maximal order}{nfcertify$(\var{nf})$} 59\endindentedkeys 60\subsec{nf members:} 61\beginindentedkeys 62\li{a monic $F\in \ZZ[X]$ defining $K$}{\var{nf}.pol} 63\li{number of real/complex places}{\var{nf}.r1/r2/sign} 64\li{discriminant of \var{nf}}{\var{nf}.disc} 65\li{primes ramified in \var{nf}}{\var{nf}.p} 66\li{$T_2$ matrix}{\var{nf}.t2} 67\li{complex roots of $F$}{\var{nf}.roots} 68\li{integral basis of $\ZZ_K$ as powers of $\theta$}{\var{nf}.zk} 69\li{different/codifferent}{\var{nf}.diff{\rm, }\var{nf}.codiff} 70\li{index $[\ZZ_K:\ZZ[X]/(F)]$}{\var{nf}.index} 71\endindentedkeys 72\li{recompute \var{nf}\ using current precision}{nfnewprec$(nf)$} 73\li{init relative \var{rnf} $L = K[Y]/(g)$}{rnfinit$(\var{nf},g)$} 74% 75\li{init \var{bnf} structure}{bnfinit$(f, 1)$} 76\subsec{bnf members: {\rm same as \var{nf}, plus}} 77\beginindentedkeys 78\li{underlying \var{nf}}{\var{bnf}.nf} 79\li{class group, regulator}{\var{bnf}.clgp, \var{bnf}.reg} 80\li{fundamental/torsion units}{\var{bnf}.fu{\rm, }\var{bnf}.tu} 81\endindentedkeys 82\li{add $S$-class group and units, yield \var{bnf}S}{bnfsunit$(\var{bnf},S)$} 83\newcolumn 84 85\li{init class field structure \var{bnr}}{bnrinit$(\var{bnf},m,\{\fl\})$} 86% 87\subsec{bnr members: {\rm same as \var{bnf}, plus}} 88\beginindentedkeys 89\li{underlying \var{bnf}}{\var{bnr}.bnf} 90\li{big ideal structure}{\var{bnr}.bid} 91\li{modulus $m$}{\var{bnr}.mod} 92\li{structure of $(\ZZ_K/m)^*$}{\var{bnr}.zkst} 93\endindentedkeys 94 95\smallskip 96\section{Fields, subfields, embeddings} 97\subsec{Defining polynomials, embeddings} 98\li{smallest poly defining $f=0$ (slow)}{polredabs$(f,\{\fl\})$} 99\li{small poly defining $f=0$ (fast)}{polredbest$(f,\{\fl\})$} 100\li{random Tschirnhausen transform of $f$}{poltschirnhaus$(f)$} 101\li{$\QQ[t]/(f) \subset \QQ[t]/(g)$ ? Isomorphic?} 102 {nfisincl$(f,g)$, \kbd{nfisisom}} 103\li{reverse polmod $a=A(t)\mod T(t)$}{modreverse$(a)$} 104\li{compositum of $\QQ[t]/(f)$, $\QQ[t]/(g)$}{polcompositum$(f,g,\{\fl\})$} 105\li{compositum of $K[t]/(f)$, $K[t]/(g)$}{nfcompositum$(\var{nf}, f,g,\{\fl\})$} 106\li{splitting field of $K$ (degree divides $d$)} 107 {nfsplitting$(\var{nf},\{d\})$} 108\li{signs of real embeddings of $x$}{nfeltsign$(\var{nf},x,\{pl\})$} 109\li{complex embeddings of $x$}{nfeltembed$(\var{nf},x,\{pl\})$} 110\li{$T\in K[t]$, \# of real roots of $\sigma(T)\in\R[t]$}{nfpolsturm$(\var{nf},T,\{pl\})$} 111 112\smallskip 113\subsec{Subfields, polynomial factorization} 114\li{subfields (of degree $d$) of \var{nf}}{nfsubfields$(\var{nf},\{d\})$} 115\li{maximal subfields of \var{nf}}{nfsubfieldsmax$(\var{nf})$} 116\li{maximal CM subfield of \var{nf}}{nfsubfieldscm$(\var{nf})$} 117\li{$d$-th degree subfield of $\QQ(\zeta_n)$} {polsubcyclo$(n,d,\{v\})$} 118\li{roots of unity in \var{nf}}{nfrootsof1$(\var{nf}\,)$} 119\li{roots of $g$ belonging to \var{nf}}{nfroots$(\var{nf},g)$} 120\li{factor $g$ in \var{nf}}{nffactor$(\var{nf},g)$} 121 122\smallskip 123\subsec{Linear and algebraic relations} 124\li{poly of degree $\le k$ with root $x\in\CC$}{algdep$(x,k)$} 125\li{alg. dep. with pol.~coeffs for series $s$}{seralgdep$(s,x,y)$} 126\li{small linear rel.\ on coords of vector $x$}{lindep$(x)$} 127 128\section{Basic Number Field Arithmetic (nf)} 129Number field elements are \typ{INT}, \typ{FRAC}, \typ{POL}, \typ{POLMOD}, or 130\typ{COL} (on integral basis \kbd{\var{nf}.zk}). 131\smallskip 132\subsec{Basic operations} 133\li{$x+y$}{nfeltadd$(\var{nf},x,y)$} 134\li{$x\times y$}{nfeltmul$(\var{nf},x,y)$} 135\li{$x^n$, $n\in \ZZ$}{nfeltpow$(\var{nf},x,n)$} 136\li{$x / y$}{nfeltdiv$(\var{nf},x,y)$} 137\li{$q = x$\kbd{\bs/}$y := $\kbd{round}$(x/y)$}{nfeltdiveuc$(\var{nf},x,y)$} 138\li{$r = x$\kbd{\%}$y := x - (x$\kbd{\bs/}$y)y$}{nfeltmod$(\var{nf},x,y)$} 139\li{\dots $[q,r]$ as above}{nfeltdivrem$(\var{nf},x,y)$} 140\li{reduce $x$ modulo ideal $A$}{nfeltreduce$(\var{nf},x,A)$} 141\li{absolute trace $\text{Tr}_{K/\QQ} (x)$}{nfelttrace$(\var{nf},x)$} 142\li{absolute norm $\text{N}_{K/\QQ} (x)$}{nfeltnorm$(\var{nf},x)$} 143 144\smallskip 145\subsec{Multiplicative structure of $K^*$; $K^*/(K^*)^n$} 146\li{valuation $v_\p(x)$}{nfeltval$(\var{nf},x,\p)$} 147\li{\dots write $x = \pi^{v_\p(x)} y$}{nfeltval$(\var{nf},x,\p,\&y)$} 148\li{quadratic Hilbert symbol (at $\p$)} 149 {nfhilbert$(\var{nf},a,b,\{\p\})$} 150\li{$b$ such that $x b^n = v$ is small}{idealredmodpower$(\var{nf},x,n)$} 151 152\smallskip 153\subsec{Maximal order and discriminant} 154\li{integral basis of field $\QQ[x]/(f)$}{nfbasis$(f)$} 155\li{field discriminant of $\QQ[x]/(f)$}{nfdisc$(f)$} 156\li{\dots and factorization}{nfdiscfactors$(f)$} 157\li{express $x$ on integer basis}{nfalgtobasis$(\var{nf},x)$} 158\li{express element\ $x$ as a polmod}{nfbasistoalg$(\var{nf},x)$} 159 160\smallskip 161\subsec{Dedekind Zeta Function $\zeta_K$, Hecke $L$ series} 162$R = [c,w,h]$ in initialization means we restrict $s\in \CC$ 163to domain $|\Re(s)-c| < w$, $|\Im(s)| < h$; $R = [w,h]$ encodes $[1/2,w,h]$ 164and $[h]$ encodes $R = [1/2,0,h]$ (critical line up to height $h$).\hfil\break 165\li{$\zeta_K$ as Dirichlet series, $N(I)<b$}{dirzetak$(\var{nf},b)$} 166\li{init $\zeta_K^{(k)}(s)$ for $k \leq n$} 167 {L = lfuninit$(\var{bnf}, R, \{n = 0\})$} 168\li{compute $\zeta_K(s)$ ($n$-th derivative)}{lfun$(L, s, \{n=0\})$} 169\li{compute $\Lambda_K(s)$ ($n$-th derivative)}{lfunlambda$(L, s, \{n=0\})$} 170\smallskip 171 172\li{init $L_K^{(k)}(s, \chi)$ for $k \leq n$} 173 {L = lfuninit$([\var{bnr},\var{chi}], R, \{n = 0\})$} 174\li{compute $L_K(s, \chi)$ ($n$-th derivative)}{lfun$(L, s, \{n\})$} 175\li{Artin root number of $K$}{bnrrootnumber$(\var{bnr},\var{chi},\{\fl\})$} 176\li{$L(1,\chi)$, for all $\chi$ trivial on $H$} 177 {bnrL1$(\var{bnr},\{H\},\{\fl\})$} 178 179\section{Class Groups \& Units (bnf, bnr)} 180Class field theory data $a_1,\{a_2\}$ is usually \var{bnr} (ray class field), 181$\var{bnr},H$ (congruence subgroup) or $\var{bnr},\chi$ (character on 182\kbd{bnr.clgp}). Any of these define a unique abelian extension of $K$. 183 184\li{units / $S$-units}{bnfunits$(\var{bnf},\{S\})$} 185\li{remove GRH assumption from \var{bnf}}{bnfcertify$(\var{bnf})$} 186\li{expo.~of ideal $x$ on class gp}{bnfisprincipal$(\var{bnf},x,\{\fl\})$} 187\li{expo.~of ideal $x$ on ray class gp}{bnrisprincipal$(\var{bnr},x,\{\fl\})$} 188\li{expo.~of $x$ on fund.~units}{bnfisunit$(\var{bnf},x)$} 189\li{\dots on $S$-units, $U$ is \kbd{bnfunits}$(\var{bnf},S)$} 190 {bnfisunit$(\var{bnfs},x,U)$} 191\li{signs of real embeddings of \kbd{\var{bnf}.fu}}{bnfsignunit$(\var{bnf})$} 192\li{narrow class group}{bnfnarrow$(\var{bnf})$} 193 194\smallskip 195\subsec{Class Field Theory} 196\li{ray class number for modulus $m$}{bnrclassno$(\var{bnf},m)$} 197\li{discriminant of class field}{bnrdisc$(a_1,\{a_2\})$} 198\li{ray class numbers, $l$ list of moduli}{bnrclassnolist$(\var{bnf},l)$} 199\li{discriminants of class fields}{bnrdisclist$(\var{bnf},l,\{arch\},\{\fl\})$} 200\li{decode output from \kbd{bnrdisclist}}{bnfdecodemodule$(\var{nf},fa)$} 201\li{is modulus the conductor?}{bnrisconductor$(a_1,\{a_2\})$} 202\li{is class field $(\var{bnr},H)$ Galois over $K^G$} 203 {bnrisgalois$(\var{bnr},G,H)$} 204\li{action of automorphism on \kbd{bnr.gen}} 205 {bnrgaloismatrix$(\var{bnr},\var{aut})$} 206\li{apply \kbd{bnrgaloismatrix} $M$ to $H$} 207 {bnrgaloisapply$(\var{bnr},M,H)$} 208\li{characters on \kbd{bnr.clgp} s.t. $\chi(g_i) = e(v_i)$} 209 {bnrchar$(\var{bnr},g,\{v\})$} 210\li{conductor of character $\chi$}{bnrconductor$(\var{bnr},\var{chi})$} 211\li{conductor of extension}{bnrconductor$(a_1,\{a_2\},\{\fl\})$} 212\li{conductor of extension $K[Y]/(g)$}{rnfconductor$(\var{bnf},g)$} 213\li{canonical projection $\text{Cl}_F\to\text{Cl}_f$, $f\mid F$}{bnrmap} 214\li{Artin group of extension $K[Y]/(g)$}{rnfnormgroup$(\var{bnr},g)$} 215\li{subgroups of \var{bnr}, index $<=b$}{subgrouplist$(\var{bnr},b,\{\fl\})$} 216\li{class field defined by $H < \text{Cl}_f$}{bnrclassfield$(\var{bnr},H)$} 217\li{\dots low level equivalent, prime degree}{rnfkummer$(\var{bnr},H)$} 218\li{same, using Stark units (real field)}{bnrstark$(\var{bnr},sub,\{\fl\})$} 219\li{is $a$ an $n$-th power in $K_v$ ?}{nfislocalpower$(\var{nf},v,a,n)$} 220\li{cyclic $L/K$ satisf. local conditions} 221 {nfgrunwaldwang$(\var{nf},P,D,\var{pl})$} 222\shortcopyrightnotice 223\newcolumn 224\subsec{Logarithmic class group} 225\li{logarithmic $\ell$-class group}{bnflog$(\var{bnf},\ell)$} 226\li{$[\tilde{e}(F_v/\Q_p),\tilde{f}(F_v/\Q_p)]$} 227 {bnflogef$(\var{bnf},\var{pr})$} 228\li{$\exp \deg_F(A)$}{bnflogdegree$(\var{bnf}, A, \ell)$} 229\li{is $\ell$-extension $L/K$ locally cyclotomic}{rnfislocalcyclo$(\var{rnf})$} 230 231\section{Ideals: {\rm elements, primes, or matrix of generators in HNF}} 232\li{is $id$ an ideal in \var{nf} ?}{nfisideal$(\var{nf},id)$} 233\li{is $x$ principal in \var{bnf} ?}{bnfisprincipal$(\var{bnf},x)$} 234\li{give $[a,b]$, s.t.~ $a\ZZ_K+b\ZZ_K = x$}{idealtwoelt$(\var{nf},x,\{a\})$} 235\li{put ideal $a$ ($a\ZZ_K+b\ZZ_K$) in HNF form}{idealhnf$(\var{nf},a,\{b\})$} 236\li{norm of ideal $x$}{idealnorm$(\var{nf},x)$} 237\li{minimum of ideal $x$ (direction $v$)}{idealmin$(\var{nf},x,v)$} 238\li{LLL-reduce the ideal $x$ (direction $v$)}{idealred$(\var{nf},x,\{v\})$} 239 240\smallskip 241\subsec{Ideal Operations} 242\li{add ideals $x$ and $y$}{idealadd$(\var{nf},x,y)$} 243\li{multiply ideals $x$ and $y$}{idealmul$(\var{nf},x,y,\{\fl\})$} 244\li{intersection of ideal $x$ with $\Q$}{idealdown$(\var{nf},x)$} 245\li{intersection of ideals $x$ and $y$}{idealintersect$(\var{nf},x,y,\{\fl\})$} 246\li{$n$-th power of ideal $x$}{idealpow$(\var{nf},x,n,\{\fl\})$} 247\li{inverse of ideal $x$}{idealinv$(\var{nf},x)$} 248\li{divide ideal $x$ by $y$}{idealdiv$(\var{nf},x,y,\{\fl\})$} 249\li{Find $(a,b)\in x\times y$, $a+b=1$}{idealaddtoone$(\var{nf},x,\{y\})$} 250\li{coprime integral $A,B$ such that $x=A/B$}{idealnumden$(\var{nf},x)$} 251 252\smallskip 253\subsec{Primes and Multiplicative Structure} 254\li{check whether $x$ is a maximal ideal}{idealismaximal$(\var{nf},x)$} 255\li{factor ideal $x$ in $\ZZ_K$}{idealfactor$(\var{nf},x)$} 256\li{expand ideal factorization in $K$}{idealfactorback$(\var{nf},f,\{e\})$} 257\li{is ideal $A$ an $n$-th power ?}{idealispower$(\var{nf},A,n)$} 258\li{expand elt factorization in $K$}{nffactorback$(\var{nf},f,\{e\})$} 259\li{decomposition of prime $p$ in $\ZZ_K$}{idealprimedec$(\var{nf},p)$} 260\li{valuation of $x$ at prime ideal \var{pr}}{idealval$(\var{nf},x,\var{pr})$} 261\li{weak approximation theorem in \var{nf}}{idealchinese$(\var{nf},x,y)$} 262\li{$a\in K$, s.t. $v_{\p}(a) = v_{\p}(x)$ if 263 $v_{\p}(x)\neq 0$} 264 {idealappr$(\var{nf},x)$} 265\li{$a\in K$ such that $(a\cdot x, y) = 1$}{idealcoprime$(\var{nf},x,y)$} 266\li{give $bid=$structure of $(\ZZ_K/id)^*$}{idealstar$(\var{nf},id,\{\fl\})$} 267\li{structure of $(1+\p) / (1+\p^k)$} 268 {idealprincipalunits$(\var{nf},\var{pr},k)$} 269\li{discrete log of $x$ in $(\ZZ_K/bid)^*$}{ideallog$(\var{nf},x,bid)$} 270\li{\kbd{idealstar} of all ideals of norm $\le b$}{ideallist$(\var{nf},b,\{\fl\})$} 271\li{add Archimedean places}{ideallistarch$(\var{nf},b,\{ar\},\{\fl\})$} 272 273\li{init \kbd{modpr} structure}{nfmodprinit$(\var{nf},\var{pr},\{v\})$} 274\li{project $t$ to $\ZZ_K/\var{pr}$}{nfmodpr$(\var{nf},t,\var{modpr})$} 275\li{lift from $\ZZ_K/\var{pr}$}{nfmodprlift$(\var{nf},t,\var{modpr})$} 276 277\section{Galois theory over $\QQ$} 278\li{conjugates of a root $\theta$ of \var{nf}}{nfgaloisconj$(\var{nf},\{\fl\})$} 279\li{apply Galois automorphism $s$ to $x$}{nfgaloisapply$(\var{nf},s,x)$} 280\li{Galois group of field $\QQ[x]/(f)$}{polgalois$(f)$} 281\li{initializes a Galois group structure $G$}{galoisinit$(\var{pol},\{den\})$} 282\li{character table of $G$}{galoischartable$(G)$} 283\li{conjugacy classes of $G$}{galoisconjclasses$(G)$} 284\li{$\det(1 - \rho(g)T)$, $\chi$ character of $\rho$} 285 {galoischarpoly$(G,\chi,\{o\})$} 286\li{$\det(\rho(g))$, $\chi$ character of $\rho$} 287 {galoischardet$(G,\chi,\{o\})$} 288\li{action of $p$ in nfgaloisconj form}{galoispermtopol$(G,\{p\})$} 289\li{identify as abstract group}{galoisidentify$(G)$} 290\li{export a group for GAP/MAGMA}{galoisexport$(G,\{\fl\})$} 291\li{subgroups of the Galois group $G$}{galoissubgroups$(G)$} 292\li{is subgroup $H$ normal?}{galoisisnormal$(G,H)$} 293 294\newcolumn 295\title{\TITLE} 296\centerline{(PARI-GP version \PARIversion)} 297 298\medskip 299 300\li{subfields from subgroups}{galoissubfields$(G,\{\fl\},\{v\})$} 301\li{fixed field}{galoisfixedfield$(G,\var{perm},\{\fl\},\{v\})$} 302\li{Frobenius at maximal ideal $P$}{idealfrobenius$(\var{nf},G,P)$} 303\li{ramification groups at $P$}{idealramgroups$(\var{nf},G,P)$} 304\li{is $G$ abelian?}{galoisisabelian$(G,\{\fl\})$} 305\li{abelian number fields/$\QQ$}{galoissubcyclo(N,H,\{\fl\},\{v\})} 306 307\subsec{The \kbd{galpol} package} 308\li{query the package: polynomial}{galoisgetpol(a,b,\{s\})} 309\li{\dots : permutation group}{galoisgetgroup(a,{b})} 310\li{\dots : group description}{galoisgetname(a,b)} 311 312\section{Relative Number Fields (rnf)} 313Extension $L/K$ is defined by $T\in K[x]$. 314\hfill\break 315% 316\li{absolute equation of $L$}{rnfequation$(\var{nf},T,\{\fl\})$} 317\li{is $L/K$ abelian?}{rnfisabelian$(\var{nf},T)$} 318\li{relative {\tt nfalgtobasis}}{rnfalgtobasis$(\var{rnf},x)$} 319\li{relative {\tt nfbasistoalg}}{rnfbasistoalg$(\var{rnf},x)$} 320\li{relative {\tt idealhnf}}{rnfidealhnf$(\var{rnf},x)$} 321\li{relative {\tt idealmul}}{rnfidealmul$(\var{rnf},x,y)$} 322\li{relative {\tt idealtwoelt}}{rnfidealtwoelt$(\var{rnf},x)$} 323 324\smallskip 325\subsec{Lifts and Push-downs} 326\li{absolute $\rightarrow$ relative representation for $x$} 327 {rnfeltabstorel$(\var{rnf},x)$} 328\li{relative $\rightarrow$ absolute representation for $x$} 329 {rnfeltreltoabs$(\var{rnf},x)$} 330\li{lift $x$ to the relative field}{rnfeltup$(\var{rnf},x)$} 331\li{push $x$ down to the base field}{rnfeltdown$(\var{rnf},x)$} 332\leavevmode idem for $x$ ideal: 333\kbd{$($rnfideal$)$reltoabs}, \kbd{abstorel}, \kbd{up}, \kbd{down}\hfill 334 335\smallskip 336\subsec{Norms and Trace} 337\li{relative norm of element $x\in L$}{rnfeltnorm$(\var{rnf},x)$} 338\li{relative trace of element $x\in L$}{rnfelttrace$(\var{rnf},x)$} 339\li{absolute norm of ideal $x$}{rnfidealnormabs$(\var{rnf},x)$} 340\li{relative norm of ideal $x$}{rnfidealnormrel$(\var{rnf},x)$} 341\li{solutions of $N_{K/\QQ}(y)=x\in \ZZ$}{bnfisintnorm$(\var{bnf},x)$} 342\li{is $x\in\QQ$ a norm from $K$?}{bnfisnorm$(\var{bnf},x,\{\fl\})$} 343\li{initialize $T$ for norm eq.~solver}{rnfisnorminit$(K,pol,\{\fl\})$} 344\li{is $a\in K$ a norm from $L$?}{rnfisnorm$(T,a,\{\fl\})$} 345\li{initialize $t$ for Thue equation solver}{thueinit$(f)$} 346\li{solve Thue equation $f(x,y)=a$}{thue$(t,a,\{sol\})$} 347\li{characteristic poly.\ of $a$ mod $T$}{rnfcharpoly$(\var{nf},T,a,\{v\})$} 348 349\smallskip 350\subsec{Factorization} 351\li{factor ideal $x$ in $L$}{rnfidealfactor$(\var{rnf},x)$} 352\li{$[S,T] \colon T_{i,j} \mid S_i$; $S$ primes of $K$ above $p$} 353 {rnfidealprimedec$(\var{rnf},p)$} 354 355\smallskip 356\subsec{Maximal order $\ZZ_L$ as a $\ZZ_K$-module} 357\li{relative {\tt polredbest}}{rnfpolredbest$(\var{nf},T)$} 358\li{relative {\tt polredabs}}{rnfpolredabs$(\var{nf},T)$} 359\li{relative Dedekind criterion, prime $pr$}{rnfdedekind$(\var{nf},T,pr)$} 360\li{discriminant of relative extension}{rnfdisc$(\var{nf},T)$} 361\li{pseudo-basis of $\ZZ_L$}{rnfpseudobasis$(\var{nf},T)$} 362 363\smallskip 364\subsec{General $\ZZ_K$-modules: 365 {\rm $M = [{\rm matrix}, {\rm vec.~of~ideals}] \subset L$}} 366\li{relative HNF / SNF}{nfhnf$(\var{nf},M)${\rm, }nfsnf} 367\li{multiple of $\det M$}{nfdetint$(\var{nf},M)$} 368\li{HNF of $M$ where $d = \kbd{nfdetint}(M)$}{nfhnfmod$(x,d)$} 369\li{reduced basis for $M$}{rnflllgram$(\var{nf},T,M)$} 370\li{determinant of pseudo-matrix $M$}{rnfdet$(\var{nf},M)$} 371\li{Steinitz class of $M$}{rnfsteinitz$(\var{nf},M)$} 372\newcolumn 373 374 375\li{$\ZZ_K$-basis of $M$ if $\ZZ_K$-free, or $0$}{rnfhnfbasis$(\var{bnf},M)$} 376\li{$n$-basis of $M$, or $(n+1)$-generating set}{rnfbasis$(\var{bnf},M)$} 377\li{is $M$ a free $\ZZ_K$-module?}{rnfisfree$(\var{bnf},M)$} 378 379\section{Associative Algebras} 380$A$ is a general associative algebra given by a multiplication table \var{mt} 381(over $\QQ$ or $\FF_p$); represented by \var{al} from \kbd{algtableinit}. 382 383\li{create \var{al} from \var{mt} (over $\FF_p$)} 384 {algtableinit$(\var{mt},\{p=0\})$} 385\li{group algebra $\QQ[G]$ (or $\FF_p[G]$)}{alggroup$(G,\{p = 0\})$} 386\li{center of group algebra}{alggroupcenter$(G,\{p = 0\})$} 387 388\smallskip 389\subsec{Properties} 390\li{is $(\var{mt},p)$ OK for algtableinit?} 391 {algisassociative$(\var{mt},\{p=0\})$} 392\li{multiplication table \var{mt}}{algmultable$(\var{al})$} 393\li{dimension of $A$ over prime subfield}{algdim$(\var{al})$} 394\li{characteristic of $A$}{algchar$(\var{al})$} 395\li{is $A$ commutative?}{algiscommutative$(\var{al})$} 396\li{is $A$ simple?}{algissimple$(\var{al})$} 397\li{is $A$ semi-simple?}{algissemisimple$(\var{al})$} 398\li{center of $A$}{algcenter$(\var{al})$} 399\li{Jacobson radical of $A$}{algradical$(\var{al})$} 400\li{radical $J$ and simple factors of $A/J$}{algsimpledec$(\var{al})$} 401 402\smallskip 403\subsec{Operations on algebras} 404\li{create $A/I$, $I$ two-sided ideal}{algquotient$(\var{al},I)$} 405\li{create $A_1\otimes A_2$}{algtensor$(\var{al1}, \var{al2})$} 406\li{create subalgebra from basis $B$}{algsubalg$(\var{al}, B)$} 407\li{quotients by ortho. central idempotents $e$} 408 {algcentralproj$(\var{al}, e)$} 409\li{isomorphic alg. with integral mult. table}{algmakeintegral(\var{mt})} 410\li{prime subalgebra of semi-simple $A$ over $\FF_p$} 411 {algprimesubalg$(\var{al})$} 412\li{find isomorphism~$A\cong M_d(\FF_q)$}{algsplit(\var{al})} 413 414\smallskip 415\subsec{Operations on lattices in algebras} 416\li{lattice generated by cols. of $M$}{alglathnf$(\var{al},M)$} 417\li{\dots by the products~$xy$, $x\in lat1$, $y\in lat2$}{alglatmul$(\var{al},\var{lat1},\var{lat2})$} 418\li{sum $lat1+lat2$ of the lattices}{alglatadd$(\var{al},\var{lat1},\var{lat2})$} 419\li{intersection $lat1\cap lat2$}{alglatinter$(\var{al},\var{lat1},\var{lat2})$} 420\li{test~$lat1\subset lat2$}{alglatsubset$(\var{al},\var{lat1},\var{lat2})$} 421\li{generalized index~$(lat2:lat1)$}{alglatindex$(\var{al},\var{lat1},\var{lat2})$} 422\li{$\{x\in al\mid x\cdot lat1\subset lat2\}$}{alglatlefttransporter$(\var{al},\var{lat1},\var{lat2})$} 423\li{$\{x\in al\mid lat1\cdot x\subset lat2\}$}{alglatrighttransporter$(\var{al},\var{lat1},\var{lat2})$} 424\li{test~$x\in lat$ (set~$c =$ coord. of~$x$)}{alglatcontains$(\var{al},\var{lat},x,\{\& c\})$} 425\li{element of~$lat$ with coordinates~$c$}{alglatelement$(\var{al},\var{lat},c)$} 426\subsec{Operations on elements} 427\li{$a+b$, $a-b$, $-a$}{algadd$(\var{al},a,b)${\rm, }algsub{\rm, }algneg} 428\li{$a\times b$, $a^2$}{algmul$(\var{al},a,b)${\rm, }algsqr} 429\li{$a^n$, $a^{-1}$}{algpow$(\var{al},a,n)${\rm, }alginv} 430\li{is $x$ invertible ? (then set $z=x^{-1}$)}{algisinv$(\var{al},x,\{\&z\})$} 431\li{find $z$ such that $x\times z = y$}{algdivl$(\var{al},x,y)$} 432\li{find $z$ such that $z\times x = y$}{algdivr$(\var{al},x,y)$} 433\li{does $z$ s.t. $x\times z = y$ exist? (set it)} 434 {algisdivl$(\var{al},x,y,\{\&z\})$} 435\li{matrix of $v\mapsto x\cdot v$}{algtomatrix$(\var{al}, x)$} 436\li{absolute norm}{algnorm$(\var{al},x)$} 437\li{absolute trace}{algtrace$(\var{al},x)$} 438\li{absolute char. polynomial}{algcharpoly$(\var{al},x)$} 439\li{given $a\in A$ and polynomial $T$, return $T(a)$} 440 {algpoleval$(\var{al},T,a)$} 441\li{random element in a box}{algrandom$(\var{al}, b)$} 442\vfill 443\copyrightnotice 444\newcolumn 445 446\section{Central Simple Algebras} 447$A$ is a central simple algebra over a number field $K$; represented by 448\var{al} from \kbd{alginit}; $K$ is given by a \var{nf} structure. 449 450\li{create CSA from data} 451 {alginit$(B,C,\{v\},\{maxord=1\})$} 452\beginindentedkeys 453 \li{multiplication table over $K$}{$B = K${\rm, }$C = \var{mt}$} 454 \li{cyclic algebra $(L/K,\sigma,b)$} 455 {$B = \var{rnf}${\rm, }$C = [\var{sigma},b]$} 456 \li{quaternion algebra $(a,b)_K$}{$B = K$, $C = [a,b]$} 457 \li{matrix algebra $M_d(K)$}{$B = K$, $C = d$} 458 \li{local Hasse invariants over $K$} 459 {$B = K$, $C = [d, [\var{PR}, \var{HF}], \var{HI}]$} 460\endindentedkeys 461 462\smallskip 463\subsec{Properties} 464\li{type of \var{al} (\var{mt}, CSA)}{algtype$(\var{al})$} 465\li{dimension of $A$ over~$\QQ$}{algdim$(\var{al},1)$} 466\li{dimension of \var{al} over its center~$K$}{algdim$(\var{al})$} 467\li{degree of $A$ ($=\sqrt{\dim_K A}$)}{algdegree$(\var{al})$} 468\li{\var{al} a cyclic algebra $(L/K,\sigma,b)$; return $\sigma$} 469 {algaut$(\var{al})$} 470\li{\dots return $b$}{algb$(\var{al})$} 471\li{\dots return $L/K$, as an \var{rnf}} 472 {algsplittingfield$(\var{al})$} 473\li{split $A$ over an extension of $K$}{algsplittingdata$(\var{al})$} 474\li{splitting field of $A$ as an \var{rnf} over center} 475 {algsplittingfield$(\var{al})$} 476\li{multiplication table over center}{algrelmultable$(\var{al})$} 477\li{places of $K$ at which $A$ ramifies}{algramifiedplaces$(\var{al})$} 478\li{Hasse invariants at finite places of $K$}{alghassef$(\var{al})$} 479\li{Hasse invariants at infinite places of $K$}{alghassei$(\var{al})$} 480\li{Hasse invariant at place $v$}{alghasse$(\var{al},v)$} 481\li{index of $A$ over $K$ (at place $v$)}{algindex$(\var{al},\{v\})$} 482\li{is \var{al} a division algebra? (at place $v$)} 483 {algisdivision$(\var{al},\{v\})$} 484\li{is $A$ ramified? (at place $v$)}{algisramified$(\var{al},\{v\})$} 485\li{is $A$ split? (at place $v$)}{algissplit$(\var{al},\{v\})$} 486 487\smallskip 488\subsec{Operations on elements} 489\li{reduced norm}{algnorm$(\var{al},x)$} 490\li{reduced trace}{algtrace$(\var{al},x)$} 491\li{reduced char. polynomial}{algcharpoly$(\var{al},x)$} 492\li{express $x$ on integral basis}{algalgtobasis$(\var{al},x)$} 493\li{convert $x$ to algebraic form}{algbasistoalg$(\var{al},x)$} 494\li{map $x\in A$ to $M_d(L)$, $L$ split. field} {algtomatrix$(\var{al},x)$} 495 496\smallskip 497\subsec{Orders} 498\li{$\ZZ$-basis of order ${\cal O}_0$}{algbasis$(\var{al})$} 499\li{discriminant of order ${\cal O}_0$}{algdisc$(\var{al})$} 500\li{$\ZZ$-basis of natural order in terms ${\cal O}_0$'s basis} 501 {alginvbasis$(\var{al})$} 502 503\vfill 504\copyrightnotice 505\bye 506