虚二次域例两（-5/50）

lilianjie

Q5:=QuadraticField(-5) ;
, t5 ~& _6 U8 D& HQ5;
5 h' z( H# W$ s: J1 O9 V
4 k& m  x% |- F( ~3 p( o# g- {( mQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
- Z! V" T9 E3 S, L: [M;
5 J; f8 {- c5 M5 ]& jNumberField(M);
% {$ @/ f3 O, u, K0 e& ^S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
/ t; P# }$ D1 J$ J" j1 t% @IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
; z3 [: b/ q4 K$ O# @! p% wIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
; }$ K  D0 D& c4 Y" u0 |0 }6 u9 SDiscriminant(Q5) ;
( I3 v" v: c" KFundamentalUnit(Q5) ;
3 @6 H1 Q! ^, ?5 a9 h4 I$ }8 h6 iFundamentalUnit(M);
Conductor(Q5) ;
' E: v/ W$ m( s6 u( }
+ ^1 a0 B0 w9 X, Y! d  wName(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
+ y7 `1 S" G$ y% |( X8 Y& e9 |6 I) WClassNumber(Q5) ;
6 G' Q+ i4 g, r( LClassNumber(M) ;
- T* j6 k7 u. \) f' u. ~PicardGroup(M) ;
PicardNumber(M) ;
& l$ O9 i& t, t3 g+ S
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
: W( Q% d! ]* @1 p3 mQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
. ~9 e/ v0 I9 F/ Z. F& d( L" LUnivariate Polynomial Ring in w over Q5
( e3 Y& o9 z" |# |; kEquation Order of conductor 1 in Q5
2 T! Z& V4 m" w, O- N$ J, p! fMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
6 n+ j% h# Q, O* i  a2 oOrder of conductor 625888888 in Q5
' k3 w' [* L9 Y2 o" l; gtrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
$ Q) b1 `$ G6 N0 T4 ttrue Maximal Equation Order of Q5
& A3 p( |8 m1 c4 o6 s7 E) r) x6 ptrue Order of conductor 1 in Q5
. q( ]) f& S( T' E9 ^$ v( z6 K, ]3 `1 t9 F; Gtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
9 }- `& z8 E. Y    <w + Q5.1, 1>
]
-20
& w, q" u% [( z0 y7 a# `; l
>> FundamentalUnit(Q5) ;
# A7 ?. E$ ~$ ?; e, _/ K; X0 F: p                  ^
! M. P* I( a5 t( t8 y2 s% bRuntime error in 'FundamentalUnit': Field must have positive discriminant


7 k' P8 _8 E0 Q' `& t7 |" z* ?4 T; @>> FundamentalUnit(M);
  K6 j2 M# M, I5 {+ N5 @                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
5 d$ M5 R9 O4 Z
20
, c/ \3 q' Z2 l. Y5 O
>> Name(M, -5);
+ X! e  x1 w1 Q7 j2 J, F5 h3 `       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

  ~2 L6 n, |3 c0 G4 @* [# H1
) ~1 Q  {( {% d- GAbelian Group isomorphic to Z/2
Defined on 1 generator
/ }7 l- d3 ~. s1 C+ x' i- v/ KRelations:
7 ^  u6 l1 L" u8 ]    2*$.1 = 0
( `; S9 w; r" S' @8 s% g# VMapping from: Abelian Group isomorphic to Z/2
* Z: I" h. b5 H* ^( P4 |* E& TDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
# y' A7 P; q6 x" tDefined on 1 generator
Relations:
: ?# |) [) K7 @. W' o$ \, I    2*$.1 = 0
6 k: R  V5 Q9 B( x3 Y$ u% ~Mapping from: Abelian Group isomorphic to Z/2
! G+ J1 M  c& n& W) Z% G6 HDefined on 1 generator
3 N) G$ Z% q% IRelations:
1 C7 v+ e" j  @2 x* c3 H+ S" _    2*$.1 = 0 to Set of ideals of M
2
/ ^. |/ J% A; r5 {3 m2
: ~0 P. l) G( Q, R, e$ O( _Abelian Group isomorphic to Z/2
Defined on 1 generator
' I( p% r8 Z( O- q6 B' sRelations:
: @1 W0 H9 ]+ Z    2*$.1 = 0
. t8 {+ d/ J& z3 I: `" O+ ]Mapping from: Abelian Group isomorphic to Z/2
! d$ P; M5 k% k. |- b$ zDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
/ |+ c1 q$ p' n( Y1 {Abelian Group isomorphic to Z/2
$ n# s0 o; g8 Y( z5 x) GDefined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
8 a& H: J" w% w4 B. I, R( _3 zDefined on 1 generator
Relations:
. R* b& d/ x: L. {# t) Y( K    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
- Q9 R  @# T& R/ R: b: |inverse]
& x( I5 V! ~* i  A# F; X; r% Q  QAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
) u4 q" P! w: A  p  SMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
+ Q5 _# U; i3 [. j$ p9 Rinverse]
4 Y, n& V/ [9 b1 Pfalse
8 w& o3 V- D! V4 Wfalse
==============
: c5 B7 I4 X8 ?2 I$ Q1 \9 J1 @

Q5:=QuadraticField(-50) ;
) J3 k2 H# Z5 GQ5;
$ T- g! Y8 _: Q' h5 E
* t$ f% [$ \+ w% Z. ?" g9 eQ<w> :=PolynomialRing(Q5);Q;
% j- n0 V* a/ OEquationOrder(Q5);
& K+ }- I- ~3 e0 lM:=MaximalOrder(Q5) ;
M;
4 T9 p* r3 A$ M1 e# HNumberField(M);
/ a' b6 E3 N7 C" |- X  mS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
$ q$ R9 `6 U3 o9 }3 TIsQuadratic(S1);
IsQuadratic(S4);
; `) m8 D0 z& V  N' V& C1 ~7 yIsQuadratic(S25);
/ M" z% y4 f! C* h- {- XIsQuadratic(S625888888);
- y% d: G3 B5 I2 JFactorization(w^2+50);  
Discriminant(Q5) ;
% V* U" P+ E9 {' c; I  p+ mFundamentalUnit(Q5) ;
' @  S7 i: K1 b$ a# yFundamentalUnit(M);
% V! B, `  ^' V, v1 {0 ?; P; J: C& QConductor(Q5) ;

. H9 D4 R3 M) \- V# w# \- ZName(M, -50);
6 h8 @1 _( u* |2 }* E" R! AConductor(M);
ClassGroup(Q5) ;
) g. [0 u6 J! p7 t2 K- ~ClassGroup(M);
+ r+ Q& y$ Z3 F- g, u4 l/ Q; YClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
- E, u/ G& X) p7 sPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
! N3 J- C; G2 a* lNormEquation(Q5, -50) ;
$ \2 r% ^# G& X3 i5 g' P* {1 Q( TNormEquation(M, -50) ;
) `) M" @/ ~: n) U) w! o" S
  J/ p- J5 q4 {! KQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
6 x- |# F/ @; r! o( C# ?/ [% t7 Strue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
1 O- f7 q' T$ k! i3 f1 ^true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
2 b0 v$ ]9 R1 W/ E" O* \true Order of conductor 1 in Q5
8 U. g4 K8 O2 \1 Rtrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
! W/ K) O& T# R6 Y! l" p    <w + 5*Q5.1, 1>
+ i# Y& _! S, ]) H- r" B]
-8
) Z2 d, t1 K2 e, d- g3 l/ X
>> FundamentalUnit(Q5) ;
: O" i) v3 h4 a; L                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

' a& l# x4 y* |0 l( [' Z* p9 A
' c9 Z2 a: O# ]2 T9 y>> FundamentalUnit(M);
$ _( m) g* K' x; ^; ], [                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
1 M( h7 }& \3 N& w4 X
8

>> Name(M, -50);
       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

" C. u" m+ P; _4 }1
Abelian Group of order 1
# Z# B- f- s2 p( J9 o* P5 eMapping from: Abelian Group of order 1 to Set of ideals of M
; r* D! T4 B9 f" e/ CAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
! _: o% s# E6 N! x4 O! R1
1
! y' E' a# z4 p  WAbelian Group of order 1
% i7 A5 i2 w6 qMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
6 P: ?! ~4 @: D8 L) D1
* m/ j( S: X5 }, uAbelian Group of order 1
: f# w% \; ]. j- O4 ~* h% dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
( P6 \$ M1 A$ j% `' ?-8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
+ d; A' n$ o4 {. [  ~2 Gfalse
false

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;
3 }/ \# c7 T' C7 E9 Y$ J
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
$ r' l1 j3 w/ D4 E- C4 R/ A; w! X1 KM:=MaximalOrder(Q5) ;
- m; x9 j; f- _2 y# z8 SM;
5 L( r7 d8 @# C% e+ h1 ]+ f0 MNumberField(M);
! c& E4 i. [4 r  jS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
% D& u6 C5 J* K6 F% T9 DIsQuadratic(S1);
' D2 y: N' a7 v" u0 p/ GIsQuadratic(S4);
; T/ `5 x, x% Y. L" v" X4 V; A( kIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
' v7 \) f; l7 f- ODiscriminant(Q5) ;
FundamentalUnit(Q5) ;
& U5 X  y9 r5 N1 U7 b( {. {7 [2 t6 dFundamentalUnit(M);
& F( h7 ^: Q' M' HConductor(Q5) ;

6 j9 W, G) C! }' q0 }3 iName(M, -1);
" T( A- O' N9 p0 @, Z, W7 `Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
- e; N# l/ N7 V1 D9 h( `+ {ClassNumber(Q5) ;
/ l3 h( a6 i9 e+ P; B/ X+ r' d: Q3 K2 rClassNumber(M) ;
" G5 N: e% N( I$ ]. R, vPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
: D; V9 U% B  w, J4 iNormEquation(Q5, -1) ;
+ y! H' P+ |& H( q; u+ a) O" jNormEquation(M, -1) ;

+ @3 c  Q% ?7 H; p3 oQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
2 u/ L2 g4 v0 S- U3 ~4 L0 H2 UUnivariate Polynomial Ring in w over Q5
" w% _: i) P9 [) e8 N5 QEquation Order of conductor 1 in Q5
/ ^# ], Q5 J2 C6 QMaximal Equation Order of Q5
) b) ?6 R* A0 M; C7 X* e& dQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
+ n" }# N" V2 s8 X; L- k1 W0 ytrue Maximal Equation Order of Q5
3 |" D$ q( {- E, u& `true Order of conductor 1 in Q5
  P3 Q) ]4 N1 h/ k0 Gtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
: L- n' B% A  ?: Z    <w + Q5.1, 1>
/ N3 S8 T1 Z3 F1 w]
; V8 O9 D2 i: X-4
6 j+ V: |/ D2 f- L; V' \; C
  B# g- L3 ^9 X1 \# v; _& c>> FundamentalUnit(Q5) ;
3 `, k( n+ n/ \; K5 G' |# ^5 m                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
  }7 p3 b, q) G

>> FundamentalUnit(M);
+ G! l4 J: U& b5 `                  ^
+ r6 l9 @/ l2 _3 ?' Y! e$ }Runtime error in 'FundamentalUnit': Field must have positive discriminant
) z0 l& I, y" E4 i
4
7 E3 _+ H2 P8 m
! K, i$ {+ i# e>> Name(M, -1);
: {/ i- K* U8 o/ [       ^
9 N9 W& `6 t; w/ k0 s; Q) HRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

; a8 h& Z0 e+ l1 _1
Abelian Group of order 1
6 c8 N2 u" g! t4 OMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
0 ^/ y$ j! w' t2 w  |0 t' CMapping from: Abelian Group of order 1 to Set of ideals of M
  B: l% _( t$ K3 J  P1
* a' t7 j! J# J) E, X0 J1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
1 k! g. I/ z; DMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 A2 {$ H8 l, m2 e3 v) |7 \& P-4 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
( G( b" t" V) d8 a# Z$ G/ C  \-4 given by a rule [no inverse]
false
false
===============
# l3 Z- k/ Y2 {; A  u: Y
Q5:=QuadraticField(-3) ;
Q5;
( }" Q% g% q# C3 z1 k
" N" G' P! i% g. ZQ<w> :=PolynomialRing(Q5);Q;
7 @* v' \7 o; w) D. n$ x+ N/ zEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
9 ]# t& E1 `; t6 T8 j. |NumberField(M);
/ F$ |8 S1 _9 B  O# K  s$ I" uS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
% n* t3 v1 Q1 \. [: k5 G4 r0 XIsQuadratic(S25);
IsQuadratic(S625888888);
: {# z2 h9 W: ^8 X+ b9 G8 g9 bFactorization(w^2+3);  
8 b, \4 U5 D# UDiscriminant(Q5) ;
4 }7 S' w! A6 pFundamentalUnit(Q5) ;
7 K0 m3 G' B! Q/ B4 VFundamentalUnit(M);
- v+ q9 I% F  Z- m, ~# SConductor(Q5) ;

Name(M, -3);
2 V6 [9 r* H7 u* k5 b- R7 D: mConductor(M);
ClassGroup(Q5) ;
7 w* z9 _) @. |7 MClassGroup(M);
( E* w4 ?% B& o) T) E% m1 U* gClassNumber(Q5) ;
* R1 N6 Y8 t( U" E" `; @ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
  ~7 }: x% V3 w0 I  b
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
" ~4 M+ H; N+ B5 ?' {NormEquation(Q5, -3) ;
NormEquation(M, -3) ;
: a; F: d$ E3 z8 L- |0 T) D
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
3 }0 k: H9 ~. [# WUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
& h  T% {# r' o/ M+ x& wMaximal Order of Q5
4 m/ _7 ?5 K" r" {9 m5 B* T) q1 qQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
4 l" j  e/ E2 _# H5 f3 ?3 H4 c# NOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
0 _& Q  v' s( M5 L8 Y" f/ \' Vtrue Order of conductor 625 in Q5
/ W' E$ m0 T( |true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
; B5 Q1 B$ j7 K+ S8 ?! @    <w + Q5.1, 1>
]
-3
, @( N* r) e9 R4 ^, m) x. R
) y% j( z0 ~: G' D) i) i>> FundamentalUnit(Q5) ;
. h/ S& Y2 H- p8 [                  ^
6 v" J# E8 ?9 k( LRuntime error in 'FundamentalUnit': Field must have positive discriminant
% ~/ J' w7 x# ?8 b6 C+ N4 j

0 R) {0 G6 m* o8 g>> FundamentalUnit(M);
- t1 I/ J" r8 B! z- }                  ^
+ n! q5 j( G8 [' VRuntime error in 'FundamentalUnit': Field must have positive discriminant
7 v$ O  N: s- p
% s( K- J, b( D- |3 n3

; v: x/ ^% X5 K- K' b9 X; {>> Name(M, -3);
" p3 z( N2 o, R0 ^, D/ r       ^
* e* e7 P/ v6 Z2 g; C; c. ^# PRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
8 j7 P9 B! B& t; l  `1 F
7 I# h2 i1 }  S( x. g1
$ v/ w/ `' J$ {Abelian Group of order 1
; F( _- g3 a! H/ X' _8 n) QMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
; j" L) x4 c( I2 _, [Mapping from: Abelian Group of order 1 to Set of ideals of M
' q; q) m2 e1 L1
) d& i3 z1 D  f, a  p6 Q: f1
6 n5 P/ X7 [& `: BAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
: H+ Q! G8 z/ R& }, }inverse]
1
7 B! v" e1 J4 r. QAbelian Group of order 1
# ?2 m) [) @* v4 ]+ hMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
Abelian Group of order 1
+ y% c, t# ?0 A4 T4 q8 EMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
/ \* e, l' i# J0 sfalse
6 Y2 m6 A9 }8 s  ?0 K( hfalse

孤寂冷逍遥

lilianjie

1 _; H/ ^( n& f, L  t
. ~/ |! ~  ?( N0 z0 z) WDirichlet character
3 k3 V2 ~# b, L5 T+ d( w' bDirichlet class number formula
( S6 j! }0 E8 `5 g0 t' Q) v6 T
虚二次域复点1，实点为0，实二次域复点0，实点为2，实二次域只有两单位根

-1时，4个单位根1，-1, i, -i,w=4,            N=4，互素（1，3），    (Z/4Z)*------->C*      χ（1mod4）=1，χ（3mod4）=-1，h=4/（2*4）*Σ[1*1+（3*（-1）]=1
+ B$ u, d8 O4 S7 t
-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
5 A* e6 m  V+ g# Oh=-6/（2*3）*Σ[1*1+（2*（-1）]=1
; E9 H/ h! S( j# _. K. R. \) m
! x- p  q" H. Y-5时  2个单位根                              N=20   N=3  互素（1，3，7,9,11,13,17,19），    (Z/5Z)*------->C*         χ（1mod20）=1，χ（3mod20）=-1， χ（7mod20）=1， χ（9mod20）=1，χ（11mod20）=1，χ（13mod20）=1，χ（17mod20）=1，χ（19mod20）=1，
# k  G* n- L4 s; ]0 M+ Q: X
+ l1 i7 J  ?+ |9 x- L
. F& F- F/ T8 `; `# t$ W7 i
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
) i1 t! S/ P& n( j* z3 b; `4 T
% u9 ?/ v; z+ t7 Y9 O
9 p* j- Q3 c8 [4 L; E  {
-50时  个单位根                          N=200
4 E4 e' B# k: N; H

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

F := QuadraticField(NextPrime(5));

+ {, B  a2 _2 G1 A& L! o' DKK := QuadraticField(7);KK;
2 G7 Q$ `! M, x$ t, ^# v: l  @K:=MaximalOrder(KK);
* E; I. p8 L, ^6 m) F  S/ w7 uConductor(KK);
" {' L$ t5 H" @2 j: dClassGroup(KK) ;
5 V- j4 u! y' Q1 WQuadraticClassGroupTwoPart(KK) ;
+ x3 f- x0 c( v) X/ ?! sNormEquation(F, 7);
6 U# w$ A8 Z1 ~5 V; l8 PA:=K!7;A;
B:=K!14;B;
Discriminant(KK)
% B5 j0 e6 F2 {" m" z; l$ b
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
2 o0 k' ?: ~; }1 h. ?3 uMapping from: Abelian Group of order 1 to Set of ideals of K
$ y( F& W7 d- X" Z6 fAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
/ i$ u( t7 `7 w( ]    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
) F2 D# \; Z) r, q( Qinverse]
( ^" A7 q2 p' u: ufalse
' m3 V: {  G# z7 ?1 ^7
' \4 |7 g/ P& W9 \6 e14
$ n# {# Z# K, t, |% E$ Z28

lilianjie

0 x6 E5 T: X0 f' v, C3 R6 H/ B' V3 l

: U  i' c, U/ `/ f# X3 y

6 ?5 Y, _7 p: t- u* \7 g4 d

/ K" W- u5 u0 h: B/ F分圆域：
9 d& j, f  _0 }8 uC:=CyclotomicField(5);C;
2 d1 P! Q, [7 HCyclotomicPolynomial(5);
9 L  s/ |1 k1 J7 d& ZC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
5 P- b$ v  W% cCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
# [$ e5 T+ H9 i6 d+ y2 y4 B/ tMinimalField(CC!7) ;
MinimalField(CC!8) ;
; I( F) T2 P& k  y- e$ b0 r* dMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
/ S' I& F1 e/ I" g. l& F. b# HRootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
( s' R" W& a9 F+ h8 t$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
0 I' h  f6 s7 T$.1^2 - $.1 + 1
8 d+ g) R$ C" V" a/ ?Cyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
  P1 f6 c! ~" _0 M1 E9 O( C: c+ s0 ]Rational Field
% \- q0 M5 r8 S' ~: [2 v: pRational Field
Rational Field
zeta_11
  u/ D7 J- C2 z( C: Izeta_111
123
7
3 b% E2 Z$ }, f" S5 ~/ z6 R4 C7
1 q: Z' W. D& \5 y! ^, H3 |1 I8 y% ~* ?Permutation group acting on a set of cardinality 6
5 W. L- I$ B/ r- y) e$ s# x' @Order = 6 = 2 * 3
2 _) L2 b) A3 B6 i4 i9 w    (1, 2)(3, 5)(4, 6)
1 I8 j) n! `; r% M    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
! y' @3 i' E  _CC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
7 s: B/ i/ a0 `. t% n& yMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

( d' P5 b* P' i, z7 o7 B
% |$ E' Y% P+ h1 Q2 y2 V分圆域：
8 `0 R; \8 t$ C5 p+ {分圆域：123

1 g! S+ z& b$ zR.<x> = Q[]
: J6 D+ [! l% d  A; H4 IF8 = factor(x^8 - 1)
  a0 b/ E) J4 ]0 r$ q" aF8
. B3 ~5 D3 j4 y( S7 d
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

5 \8 m% Y2 i6 |Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
. K2 @" a: t5 J! Q0 t" lFF:=CyclotomicPolynomial(8);FF;
  n3 p' `, `7 O2 ]2 D
# \; L, g) {8 ?6 H5 [F := QuadraticField(8);
F;
; F( P; [0 l( j( k* \$ I" B: YD:=Factorization(FF) ;D;
! j* r' c! Q8 R( hQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
4 B7 m, q6 C, z* s, HCyclotomic Field of order 8 and degree 4
$.1^4 + 1
# Z+ y( g: G9 s# v' \0 F8 [% JQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]
/ J" Y; l9 O% _2 P  d. b
' d& l3 ~: d! M& sR.<x> = QQ[]
F6 = factor(x^6 - 1)
F6
& l1 p( X- t% W3 E
' {. S! z) L/ j& O- g: j(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
! c4 u- u: Y( C' Y
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
0 i' H, _: {' L5 M2 X
7 f: q$ @( V; K- v' x# G" V) IF := QuadraticField(6);
) H* U6 m2 \9 D" a2 j0 U) n$ GF;
* T* r6 T2 e  h3 E! q& O  m7 P/ JD:=Factorization(FF) ;D;
% M# |- J8 `4 N/ s- p4 W0 sQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
( X4 N9 U: H# U# mCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
$ g  D+ R1 s5 \* x# z* }[
    <$.1^2 - $.1 + 1, 1>
' o& h9 k7 Y; d& p1 a% k/ C  R" r' f' z0 u]

5 `, J( ~- g  L( f" O3 xR.<x> = QQ[]
. N7 i& r7 T' o3 z2 kF5 = factor(x^10 - 1)
+ O1 d1 `5 G  x# m7 ~+ [( c, ]F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
0 {& C' C3 q7 Y! f1 s# ^8 G! T& o1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

3 z- N# f7 }/ S; EQ<x> := QuadraticField(10);Q;
$ T! Z) m5 N$ b& J& S7 [C:=CyclotomicField(10);C;
" u1 _4 ~" [; ?" ]: UFF:=CyclotomicPolynomial(10);FF;

& y  M, w8 s% ]: t) m4 y$ m1 nF := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
% m3 F* f+ t+ J$ {5 ^Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
0 [0 s* P% `7 a3 @% {; }$.1^4 - $.1^3 + $.1^2 - $.1 + 1
% |* z4 B' ^  R* AQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
; S0 S' _+ P: ^& g  {    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123