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标题: 虚二次域例两（-5/50） [打印本页]
作者: lilianjie    时间: 2012-1-4 17:41
标题: 虚二次域例两（-5/50）
本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 + G+ {# ~6 {3 `) P! ?

$ e8 f7 G0 A& p% H( K' h& YQ5:=QuadraticField(-5) ;3 S+ _: u& y0 h# @* O3 S
Q5;( o" k# Y% o8 W* {; D+ ~1 @
3 J* K& [/ V7 X+ n& n
Q<w> :=PolynomialRing(Q5);Q;
+ ~& b9 e2 Z) l' e4 m% W1 QEquationOrder(Q5);
& G) y1 d' I6 g# m4 VM:=MaximalOrder(Q5) ;
% X, G- ~" @: sM;* E8 ?+ X& f( D' ^
NumberField(M);; i, L, _+ T+ [" O
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;5 j" H2 o; |6 z
IsQuadratic(Q5);+ N: X. G2 N. B2 O  `+ ]% Y0 {
IsQuadratic(S1);! v7 Q! @% R) x; D1 k; k5 r
IsQuadratic(S4);4 f: ^0 \. ^# ]) I" c# T
IsQuadratic(S25);
! H7 C7 k: M( xIsQuadratic(S625888888);
* t9 ]9 A2 l4 J" e( [Factorization(w^2+5);  
$ ]0 e) X5 |% tDiscriminant(Q5) ;( Q2 w  n0 P% j/ f1 g
FundamentalUnit(Q5) ;
4 [: ]7 h/ e5 Q( oFundamentalUnit(M);3 U9 y/ a0 R, Z6 ^- A- A4 o9 D
Conductor(Q5) ;' Y4 H5 x- C8 G, a8 g% J7 B9 Q
9 o' l, q9 L# p' e* E' ]4 j3 q6 b* c
Name(M, -5);
6 Q2 }" |( H/ _9 iConductor(M);0 @6 H, W1 s! I  H$ Y
ClassGroup(Q5) ; ; _* D  u  c. X
ClassGroup(M);( u, {! B3 E  m7 P( ]
ClassNumber(Q5) ;
# i+ M) D, V: ?ClassNumber(M) ;# S5 e8 |% J( |, ]0 g- j! t
PicardGroup(M) ;! h/ g# ?6 U' F' |  o# n
PicardNumber(M) ;: T% S0 d2 h$ k
; E0 n, t1 t) h( r. ]' ~8 T
QuadraticClassGroupTwoPart(Q5);2 z! b: g, x* |; s- a; c' s
QuadraticClassGroupTwoPart(M);
" c6 s1 v5 W. M3 i" w  X) WNormEquation(Q5, -5) ;
7 k7 z1 I9 X* ~NormEquation(M, -5) ;/ D% _+ N* j+ H0 |! u8 s* h3 c
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
5 u2 K; i. I+ {4 XUnivariate Polynomial Ring in w over Q5/ S+ `& H5 D1 G4 S; J- \: w+ m
Equation Order of conductor 1 in Q5) Z" J9 w" I( m5 J) v8 l' S3 O4 @
Maximal Equation Order of Q5
  t! Q! A8 j8 lQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
4 C6 [' @. v9 L: k( k; Z" ]Order of conductor 625888888 in Q5
# D7 }2 U6 s, g; M" p7 m( {& Ltrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
- R# y# g# Y8 `! n! ftrue Maximal Equation Order of Q5
" b  |, U2 M5 l' y- I4 N7 btrue Order of conductor 1 in Q5
' v/ F: A- g+ r2 R% R8 }. Rtrue Order of conductor 1 in Q5- W2 K$ u" e1 n  U! E9 @" F
true Order of conductor 1 in Q5
, o0 |1 d% e6 H0 ]4 _[
) |; h" z5 S" c  X- P$ D% j    <w - Q5.1, 1>,
  \) x4 Z2 |/ n# x# N    <w + Q5.1, 1>
( B, Z, a4 W, i0 b]7 J9 A8 J+ }3 \7 J1 U! b" j
-20
7 d, s, X. k& Q8 L
( S- D& I9 H8 Y( N% f- W>> FundamentalUnit(Q5) ;4 ~4 N0 ^1 N8 X
                  ^. Y' @( x4 u6 W" z. T. }
Runtime error in 'FundamentalUnit': Field must have positive discriminant
, m- g1 _" P' m4 E( n: ~; i: E9 V+ X. j  L: B! F( G0 {" p! {

& |! B' L  {" R: H>> FundamentalUnit(M);
- ?' M: A- ?% V- @2 a) a: q2 o                  ^3 d) ^; k# K7 @  q( z
Runtime error in 'FundamentalUnit': Field must have positive discriminant/ C) l. \; s# |7 f9 u  ^

6 D- k7 ?) @, c% p1 U. H1 S20" I/ ~$ f2 ?- Y

8 F9 a' w5 S) f& X+ |; N$ W+ C( ^>> Name(M, -5);! d/ w; \! Q1 _4 y. z
       ^
3 H2 N# s& j( ?1 f8 i, eRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]; s) F' C" j) C6 p0 g6 R' |

0 V* ~6 B! j: R1
# l! ?4 G6 K8 l9 U  z  RAbelian Group isomorphic to Z/2
0 I. H$ l* \. n( }4 w* ]( o  bDefined on 1 generator( h/ q) X- T# D& ?& k+ I5 j
Relations:
2 R1 C4 R( \, c, L    2*$.1 = 0( o0 i1 B2 [/ j* _( K3 O
Mapping from: Abelian Group isomorphic to Z/2
7 ~# `' q; [9 @7 v7 |Defined on 1 generator
8 C% F! ?7 j  H$ p' @Relations:
7 [: Z3 C  a3 q' j. g# q- W+ ~, w    2*$.1 = 0 to Set of ideals of M
7 h3 v. v" C; U# g3 E+ qAbelian Group isomorphic to Z/26 {& i1 |: {2 L$ s
Defined on 1 generator, Y/ s; v+ V3 ~; V, X( J) @
Relations:, E$ q! q# `, m) C, g' v
    2*$.1 = 0. ?2 I! O* V7 r. L. Q
Mapping from: Abelian Group isomorphic to Z/2
' o# e  q  ]$ nDefined on 1 generator
0 A; V1 Y7 F* P4 T0 T. R0 wRelations:* C1 \( X0 u# [0 I: H
    2*$.1 = 0 to Set of ideals of M
3 _6 W: Q0 V4 X5 g) D2
8 t3 U, f8 M( g( |% y& x2/ q5 _* c0 B( F  C- W- |
Abelian Group isomorphic to Z/2) f) F0 ?$ Z, p6 W. s
Defined on 1 generator
; N7 Q* W4 v8 f3 c4 FRelations:  N- a" e9 R5 L  L
    2*$.1 = 0
- v. \& M9 |" \  h  C: {% J3 g8 ZMapping from: Abelian Group isomorphic to Z/2
: S4 y* ]& I0 n2 DDefined on 1 generator( h7 y* }* Y! t3 e: f
Relations:! u4 \0 H' f. \0 b
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]+ j. P% v( J4 W- [2 \
2" X. n7 ]( H/ D- T* A) T
Abelian Group isomorphic to Z/2
" m- Y# M0 Q8 s& ]: b1 c! hDefined on 1 generator. e# S( L  b! P3 f$ R* v8 W
Relations:) t' g/ l8 }' x4 G5 k  K
    2*$.1 = 0
7 y/ R: i% t2 }" S& P* z+ Q8 E0 UMapping from: Abelian Group isomorphic to Z/2
# T/ M( g2 B! C/ h  b' lDefined on 1 generator" l+ @1 j- n& f
Relations:
( V& C4 M! W& D/ @2 |    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
( g$ X" Q0 I. {, W; f( i  f+ yinverse]$ ?! c! i! E  M1 _5 F
Abelian Group isomorphic to Z/2
! ^/ {' C( w6 i! I0 P# UDefined on 1 generator
2 L- R& U5 y- X5 Q$ C" u, X! YRelations:/ V& n5 _+ }! f
    2*$.1 = 0+ e% y; r7 P' f* Q( N0 u
Mapping from: Abelian Group isomorphic to Z/2& Q0 x. Z) W+ E8 ^: b9 J- ^9 @, R
Defined on 1 generator
7 u( w# @8 k! E6 e. o  a4 MRelations:! K0 A# s- `" C. n, G2 Y: W
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
: y7 s. |& H# N  W2 V2 @inverse]1 j5 f( T9 T& H( C5 t7 m4 r' J7 h
false5 w4 s! Z( ~  h) |
false
/ H4 J% D& v- ]$ ~# u. a==============
1 I9 k3 F. A1 T1 L( K8 C* _: C4 D" {# @
6 i, z1 T# h" g5 s! J4 c
Q5:=QuadraticField(-50) ;
1 U, ?) a$ y+ m+ `4 P) nQ5;6 J; ^6 Y) v  D8 S
, v, b( d6 F" b; d3 k* Y1 M
Q<w> :=PolynomialRing(Q5);Q;2 K( N0 }: b+ p/ m
EquationOrder(Q5);
/ N) G$ j% e( f( i/ q& ?M:=MaximalOrder(Q5) ;
. q8 Z2 P2 C8 |& @, B  N+ d* pM;
3 p2 F0 W- l1 f8 ~+ LNumberField(M);' O1 O& w2 y% ^0 T9 k- W
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;
+ s# ^" q; A* I  p4 d) _$ rIsQuadratic(Q5);
' h/ |/ A) ]% X- sIsQuadratic(S1);
: x% q! t# n- ~* w. aIsQuadratic(S4);
6 {9 e1 `2 t9 z' z8 Y8 H9 d, yIsQuadratic(S25);
/ N9 {8 [0 |# t$ o) eIsQuadratic(S625888888);
" k  o. \% E: fFactorization(w^2+50);  
+ r% N0 D4 {8 T% b2 i; jDiscriminant(Q5) ;
- M$ j0 M1 L) y  x" R$ o9 c/ JFundamentalUnit(Q5) ;. \/ I% @  H1 U! i: |* K
FundamentalUnit(M);
' @8 z9 ]  [% m" Q, I* k# zConductor(Q5) ;/ ]% R: S) E  u7 p1 {+ [# L1 \' R9 Y

* ^: O' T- A! c# X9 o; U3 ^Name(M, -50);" l3 Y. b8 S3 l1 L8 \5 }$ H8 U+ O2 s% ]
Conductor(M);
& v$ x+ i% k/ X) gClassGroup(Q5) ;
& w: d/ B4 |  p( @# QClassGroup(M);4 ]( T( d! x. F* Y! I2 v# M( L
ClassNumber(Q5) ;+ C% ]. p* _/ D2 S, q
ClassNumber(M) ;
' D# p  I! `6 \$ O. H2 Y5 m% `PicardGroup(M) ;
, D2 M, l' Y0 K6 C1 j: cPicardNumber(M) ;
4 n3 o# s' j3 U) }" U% |8 k
( Q. [" x% ~" D4 y; FQuadraticClassGroupTwoPart(Q5);$ j6 s& }3 Y3 A, a! s
QuadraticClassGroupTwoPart(M);
1 ~1 p' k, J9 I8 S8 L% n! KNormEquation(Q5, -50) ;
/ z. I, m& @  ~3 {  @NormEquation(M, -50) ;
+ t0 [. p! |  u! H6 f
0 S7 O* b4 W2 iQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field; A( `+ O& p0 h" {6 Y  B' [  z% |
Univariate Polynomial Ring in w over Q52 p* d  ]6 n1 l7 R$ T
Equation Order of conductor 1 in Q5
  k0 M- }* s/ U( @0 tMaximal Equation Order of Q5
" @! \& I. N  kQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
2 O/ `9 m$ |4 H; b3 J6 F9 A: l8 o4 IOrder of conductor 625888888 in Q5
1 i4 ~& `" @4 n0 t, H' Z  T4 a7 Ztrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' J7 P) a# V, d( j1 F3 n+ Atrue Maximal Equation Order of Q5. ]8 F4 A+ X- }+ u5 b
true Order of conductor 1 in Q5
) G( L; g7 ^! ~9 j8 Itrue Order of conductor 1 in Q59 H0 y- X: G* J) W& _  ^# o1 G5 G
true Order of conductor 1 in Q58 @" o0 G/ r& X$ }
[0 Y& n2 H& N% w" z
    <w - 5*Q5.1, 1>,6 B9 K" N+ x: `4 i1 M: {4 ~. G$ w- |2 v
    <w + 5*Q5.1, 1>- q. |  [( t% g! b
]/ k; h, d3 J$ a! q4 M. Z
-8
6 }7 @9 X1 A! B9 d
. I6 ~* M  W- B  T! @>> FundamentalUnit(Q5) ;
; n. E. @" M0 ^1 ^. w6 L# |- Y! Q3 i                  ^% \2 q! H& l% F, t4 d( f
Runtime error in 'FundamentalUnit': Field must have positive discriminant
0 C0 n+ B5 v7 ~# d. M$ i) M# o/ |; D$ J( X. e
" a' _& T) o+ u
>> FundamentalUnit(M);* o/ o2 O0 ?& H' d" U% {. {  [; T' m1 }
                  ^/ |# R8 K: H9 a: ]/ c/ X
Runtime error in 'FundamentalUnit': Field must have positive discriminant
* b' s# ^- p$ y* g6 O
5 v: \2 ^  E# C82 v& J, F* [0 k  B
* S# g& E* p  _0 x
>> Name(M, -50);
9 P" X3 h% |! F) D! l/ N       ^
3 T. C7 W' M' m% W+ XRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
/ m! e! F/ o& \, P4 `0 X1 @
; C: W6 V+ ]5 k" K1
6 Y; Z  L, @- y7 s+ _Abelian Group of order 1
3 Z/ S8 A1 |: \: qMapping from: Abelian Group of order 1 to Set of ideals of M
/ S1 B3 R( o, Q/ [# ^6 ?& O' I* kAbelian Group of order 1( v$ U7 C$ {1 @, t+ ^9 t3 s6 t
Mapping from: Abelian Group of order 1 to Set of ideals of M+ `; L# S! h5 b( J/ X/ Z
1: p7 t' l# \  O9 y1 _# u9 F: x
1
0 R  @! G& ]: ?& F0 o8 B) X6 E. NAbelian Group of order 12 n  E% h3 D: G' L
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
2 h7 \7 N/ L+ n  `inverse]: f" L, D6 x3 }
1. C+ s3 d2 X3 D; t
Abelian Group of order 1( m$ v2 M* z. p. X) \, h- Z& W
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant8 T2 O) C6 g/ Z4 ^; [1 d; x
-8 given by a rule [no inverse]# ~& B, D5 P( w, z9 o- M" @: W8 n
Abelian Group of order 1
1 I0 ~% W' x# z# Q1 N9 g1 qMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant# k7 z$ n/ Q7 V
-8 given by a rule [no inverse]
/ x1 Y" {7 ?3 _/ a0 v2 Xfalse  T, W0 ?9 r4 X) ^* c
false
; o5 ~) |& m& c/ }1 h" H+ H. [) k4 v
作者: lilianjie    时间: 2012-1-4 17:51
看看-1.-3的两种：
) {+ Q# {( B: s8 H
6 N2 }. J# @- ~4 R( c- kQ5:=QuadraticField(-1) ;7 ]6 H' I' W4 G. k" }
Q5;
5 j, b* c1 U  s  Z
* C1 z! }  w. ?Q<w> :=PolynomialRing(Q5);Q;
! s( M, _' w1 Q9 t7 h! Y2 XEquationOrder(Q5);
* ~, r, d4 q8 }3 S2 cM:=MaximalOrder(Q5) ;6 y$ y; C; l( Z, U, ~2 M, w. v
M;* g! M* I# K3 G* z
NumberField(M);$ S6 e7 Z! s: o$ z" t' q
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;/ Y# p* _2 n& K/ M
IsQuadratic(Q5);9 O# v4 z. g# I
IsQuadratic(S1);
6 q/ x% Q1 q, Z: O- O/ e2 k/ h9 a5 RIsQuadratic(S4);( T: D, W& K( U: t8 f5 B; z4 O
IsQuadratic(S25);; E6 m# D3 P" O4 F
IsQuadratic(S625888888);
) y: q+ @8 h. _6 C3 Y; SFactorization(w^2+1);  
/ l8 O/ f: V+ t3 j8 e- |Discriminant(Q5) ;
' `7 ^4 c2 z: Z, z3 T, o  N8 yFundamentalUnit(Q5) ;) [2 t" ]8 R" u) v8 x6 D, M/ `) S
FundamentalUnit(M);
  f+ L" ^0 J* h: L2 u* pConductor(Q5) ;+ S6 M& J6 s( }' J% \

5 n) C3 R$ Q' T2 c; w3 H4 d6 eName(M, -1);" P* }- w9 m% z7 x! @
Conductor(M);
2 H. u, o& E' m* B- C( ~3 uClassGroup(Q5) ; 4 \1 X& f% f5 }# R
ClassGroup(M);
; `2 y$ ~0 |  @  f7 w/ TClassNumber(Q5) ;
; Q3 B1 m6 b2 y0 H$ L: QClassNumber(M) ;
# L) d. h* X9 E$ u- TPicardGroup(M) ;
$ E  a. b# H8 E- JPicardNumber(M) ;
! b' {8 p- ?% e% o  D. @4 r2 s9 T
5 g7 V: m& O  F2 dQuadraticClassGroupTwoPart(Q5);% |$ g  v* R& o; A* d9 s
QuadraticClassGroupTwoPart(M);2 z$ Z- s5 ?, }9 m) ]5 ~6 ~" V- W' b
NormEquation(Q5, -1) ;' ^: g1 W# E9 V1 \
NormEquation(M, -1) ;$ r# M4 Q$ i) L" d/ C

" s7 z1 {' ?4 Q- lQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field, A, v2 o) u2 \6 ]- @- m
Univariate Polynomial Ring in w over Q5
. s; h- E3 q' HEquation Order of conductor 1 in Q5
$ z+ [' r/ c$ `, F9 x9 v9 H" SMaximal Equation Order of Q5
$ \/ S% L- ~* r, L2 ZQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
- L  W1 u1 _$ tOrder of conductor 625888888 in Q58 O4 @- C% e% @4 I8 W( o" E" F, Y
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field7 j. Y' F$ [# V" v
true Maximal Equation Order of Q53 }8 ^  ~) \0 N* n! w4 @0 A& o- x1 Y
true Order of conductor 1 in Q5
$ b+ Q4 b' @0 X8 s% [- _9 @true Order of conductor 1 in Q5& S3 j/ m: |8 q' P2 E
true Order of conductor 1 in Q59 n5 m0 }# k# m5 r* b
[
/ w2 o' R6 l( X0 C" X) d9 [    <w - Q5.1, 1>,0 f, A1 R3 G. [/ v; n
    <w + Q5.1, 1>
; x# p/ O  P( {]1 G0 W& ^% n  w5 T
-4% B* ~' X0 a# i2 D

. B, I6 ~. C! a5 ]- F7 P# Y>> FundamentalUnit(Q5) ;, B  B& t5 N" P& w: k
                  ^9 ~! \  g# T% r
Runtime error in 'FundamentalUnit': Field must have positive discriminant( U3 {' ]" [+ b3 E3 R

: P5 i* \5 R4 U, e5 |4 X& k
6 U$ G! M' g" z( J' |3 a>> FundamentalUnit(M);( W' t3 M* k! ^; ^/ A6 i
                  ^- k" ?2 X. l* O8 G. z: x
Runtime error in 'FundamentalUnit': Field must have positive discriminant
4 l# L* ~3 W) s  b2 K; J
1 o" g; @% l& P" L/ w4
" `& M% Q1 G: X( ^2 k% E
; C8 S7 [0 y2 S8 O>> Name(M, -1);2 T$ n) H( J0 F4 ]$ x
       ^8 ]# w. l5 {6 a4 W  t3 P
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]. _$ `+ t: x6 Q* f

1 D2 c8 p! J% _4 W+ r) {/ p1
: r9 s! g! u* k: Q/ U  _Abelian Group of order 1
  ~# Z- @/ D& Y" z! ^2 w6 G2 wMapping from: Abelian Group of order 1 to Set of ideals of M! H8 F+ N, ^' J  y- Y! K
Abelian Group of order 1: A9 y+ A  Q5 w$ J8 E* I
Mapping from: Abelian Group of order 1 to Set of ideals of M
+ c/ Z) ]# k; t0 n/ E17 T( v( }* a3 e" N: {3 u& I
1
9 t+ l1 Y6 a. _: {: D5 |1 \! QAbelian Group of order 1
2 D/ R  v1 b$ l" L" jMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no# O2 z5 ]' C, P
inverse]
+ {7 h  ?/ O. \: k3 ^" W1
5 r+ f8 y. ^6 _, g" s& BAbelian Group of order 18 o! M- q! n: S  \2 \; B
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant, V  F  i" i- z
-4 given by a rule [no inverse]+ v6 c7 _; q2 B' M
Abelian Group of order 1
; n+ _$ M8 G- N* N. I1 |Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* B, F/ k+ d3 r-4 given by a rule [no inverse]
1 I- m3 z5 J' C! U+ m) V3 }: yfalse
7 F; g  Z& ^( _. b9 Z$ H3 pfalse
4 A# Y5 A% o' g8 W===============  U8 P5 c' `2 ^: q2 s" G1 W
( V- ^# X  r4 m8 ?
Q5:=QuadraticField(-3) ;
3 b' i  h; q  |- NQ5;
$ |2 \) \5 K8 `  H1 a/ \0 f/ S* R
1 y* K" @5 S- d' o; J9 d/ _1 T5 oQ<w> :=PolynomialRing(Q5);Q;
/ S( F) D5 |  R+ e+ VEquationOrder(Q5);: R- x. N4 y5 h+ z$ _6 T0 \0 V) w
M:=MaximalOrder(Q5) ;
# k6 b) X" m+ e! Z! ?M;  ~7 J' a1 z$ \# \. P. [& Q) R$ [5 X
NumberField(M);
. `1 j; b8 K5 ^. i" PS1:=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;
+ W' h2 E  q, S$ H- \* d" vIsQuadratic(Q5);
# I4 `: q5 ^& w% s3 y9 H7 [IsQuadratic(S1);' ]  [/ u. v* P" F0 g( p" u
IsQuadratic(S4);
: x0 v& i2 H( u3 z/ J3 @IsQuadratic(S25);* {' o! j* `! b6 ^; U
IsQuadratic(S625888888);- l2 z& ?" |" Y# s7 o4 J
Factorization(w^2+3);  
" q& }+ V1 \! \Discriminant(Q5) ;
. L) C9 ?- {/ ?+ G0 SFundamentalUnit(Q5) ;" [0 ^7 `4 I( a: I5 X
FundamentalUnit(M);' F( Z1 T1 m0 O0 y: M1 v1 F7 B
Conductor(Q5) ;
7 v& s9 J) t) V3 c9 `; ]- B( C, J- w( q+ }! a" m
Name(M, -3);
) a3 c" i& {4 h% GConductor(M);
5 b# t( K' H6 j& f% @, g: T6 X7 e0 V$ JClassGroup(Q5) ;
; z6 B+ [% H) B! K# wClassGroup(M);2 n( y  H& x' {1 a4 V4 ?$ M
ClassNumber(Q5) ;
- R' u2 K+ _; q) d+ V; hClassNumber(M) ;
2 \1 \3 ^' @6 w, e4 c) A( U! q" ~PicardGroup(M) ;
0 o% d" J) m! F! W( {" uPicardNumber(M) ;
( S( r# f& a; a7 ~; \7 d; h
& T" Y1 Q& l8 r) a6 TQuadraticClassGroupTwoPart(Q5);, {; F3 K: l, G2 c; r
QuadraticClassGroupTwoPart(M);9 E$ [, q1 a: h& E" n' H8 s
NormEquation(Q5, -3) ;
( s* J6 t7 a* O. x2 N3 s1 pNormEquation(M, -3) ;
: _3 P3 e4 w+ K8 i2 e5 }
4 [$ J4 |4 G+ l4 b1 dQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field7 d& b$ O$ j7 S, j, x3 G8 i* _
Univariate Polynomial Ring in w over Q5& }' Q! V2 w5 `+ ]% _; i! b8 V
Equation Order of conductor 2 in Q5  N4 q+ v) S" o4 v# K" w/ b) {# u" m
Maximal Order of Q5
7 B7 S6 `5 R/ K( a1 k0 \Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field9 Q7 v3 P' i( ^2 W1 D& D
Order of conductor 625888888 in Q5
' ]4 j) N" h' z4 C0 l) m7 X+ |true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field) O" f3 L' j" j: n# D0 E0 [
true Maximal Order of Q54 X& C' U2 l* ~9 f! l% A$ F
true Order of conductor 16 in Q5* R3 Q5 r4 f( g+ F0 d* Q! ^0 X
true Order of conductor 625 in Q5
( ]4 x9 E7 x: ?1 n: R$ g, n; H2 n3 o1 ttrue Order of conductor 391736900121876544 in Q5
2 Z& }" Q& S3 I; ?[
0 r9 l9 K+ \# G+ u* A- a. C. N1 v0 F: S* c    <w - Q5.1, 1>,8 p6 r5 i) s+ p% |+ a
    <w + Q5.1, 1>2 a. |3 T( R' ?
]
5 u! G4 i: x8 r, Q& A0 d-3# }7 O: r* `& m% [
" }( _! F$ g. ?) c( @/ u' D
>> FundamentalUnit(Q5) ;
% P, d- I3 d3 K  f" H& p                  ^
# U5 |+ y* s+ b) H' ^. ]Runtime error in 'FundamentalUnit': Field must have positive discriminant  A1 s, {. c1 P7 d( ?2 B2 b
" q0 t' {/ l# b  T0 o

! A3 s( Y- K* D- ]& t>> FundamentalUnit(M);
0 y, }* }" C) G) j' s0 C                  ^
+ ~: o! |* x- }; o1 O7 {4 }Runtime error in 'FundamentalUnit': Field must have positive discriminant
" p- T. k6 _- f; {# i; T5 s( A
3
2 h( F/ ~2 q0 n) e- j9 Z8 K. k& f; ]# L' O  o
>> Name(M, -3);
5 K! R' _) z3 L  D; ^       ^0 o8 @* H. ?. U- p
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]& y0 e' p) l" R
# k* |+ c+ B. t
18 T& B- Y# x$ Y( L3 C( |
Abelian Group of order 1
& R8 l1 r& P3 \# G. yMapping from: Abelian Group of order 1 to Set of ideals of M
8 `4 r: X+ v/ a2 R- fAbelian Group of order 1
5 A" P% a& e8 A8 h5 s1 K/ sMapping from: Abelian Group of order 1 to Set of ideals of M$ c+ w+ Q# d/ y5 Y, G
1# K3 P! A3 T, J. V
1
# X5 g+ M- k* y; J' \5 `. XAbelian Group of order 15 R% n9 e1 B( B2 q
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no. l! W6 m# J& O  R8 d  v1 l
inverse]
- c3 G# U& K8 P  p  B" s1
" h$ U3 ^% @- k4 JAbelian Group of order 1, T) y' p0 A* w+ w6 b
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant  R5 b+ w; ?, i# g9 Z
-3 given by a rule [no inverse]6 o7 r& M; s$ ~2 o
Abelian Group of order 1  t& ^5 c6 h" ?7 S+ R7 l8 e
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant, t5 }& n) n( ^
-3 given by a rule [no inverse]
4 q' r3 m9 ]& Ufalse
) W: e& V" _4 g% o/ Gfalse
作者: 孤寂冷逍遥    时间: 2012-1-5 08:36
作者: lilianjie    时间: 2012-1-5 13:02
本帖最后由 lilianjie 于 2012-1-5 15:20 编辑 9 ?0 g0 C+ C: Z0 _4 A! M
) `; T4 o3 C, k2 c. i  v: ~7 _
Dirichlet character
3 B8 b* y" G) ~8 ?1 u, _Dirichlet class number formula# h2 p4 D0 P2 |; [  J+ j
4 B2 {% s- }- c, W5 b/ q% B
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) ]% D# G& N2 F
' S& L& ]  R' t0 w* `  S9 a
-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
3 M6 N3 Y3 I; Q" f2 Y
9 V, C" L/ M* w% u, H, ]' H/ R-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，2 R/ i  }& k( O
h=-6/（2*3）*Σ[1*1+（2*（-1）]=16 j. v* A0 J) f7 J3 v% Z
: H1 d6 k' H$ E) i" R$ O1 Z2 a1 L
-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）=1，9 m, T+ K' F3 ^8 p% W+ [4 A3 P

) L* ~$ d4 v+ P; p& R8 d; N  ~! K6 z0 ^( q$ T; U$ J. `4 ^$ e

. K4 N8 a5 N8 u4 X6 }5 {% Nh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2  P0 C( e4 G% Y
9 z" l$ Y7 O# M( K: _3 d* |
4 }+ {; p- M# l& q
8 X" }) K3 w3 [) b! \! c
-50时  个单位根                          N=200! N1 E$ a5 }0 D3 g
作者: lilianjie1    时间: 2012-1-5 13:51
Dirichlet character

21.JPG (79.18 KB, 下载次数: 254)

21.JPG

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11.JPG

作者: lilianjie1    时间: 2012-1-5 20:37
pell   equation

11.GIF (22.03 KB, 下载次数: 261)

11.GIF

作者: lilianjie    时间: 2012-1-9 20:28
本帖最后由 lilianjie 于 2012-1-9 20:30 编辑
, \1 m' f/ }( g, q; d& ]  t7 b  R" l$ R. Z, f! n
F := QuadraticField(NextPrime(5));' P9 A( K! X- T/ }* j0 }; v3 m3 f

+ j: H0 J$ A- A6 f& m7 mKK := QuadraticField(7);KK;
/ H/ b& T( _" i8 LK:=MaximalOrder(KK);' h% L) z0 N( z- s5 I# A# N
Conductor(KK);3 V. S+ z" ~+ k, C
ClassGroup(KK) ;" ]2 N6 X* a2 M  l
QuadraticClassGroupTwoPart(KK) ;
0 d0 H% C( H9 A6 D  M8 Q& `NormEquation(F, 7);8 K; u) t! u) B4 D1 {6 ^, F
A:=K!7;A;& U& |) p3 W5 _" _- j* O( w
B:=K!14;B;
4 r4 `; Z" t2 j) q& {6 XDiscriminant(KK)- F8 R- C, r+ I; }$ ]0 k

; A2 E' f( O+ j) tQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
4 U" V: A. c8 l; g4 }5 W287 [; L( |5 b' s  E* R# L
Abelian Group of order 1
3 m" Q7 ?# Y# P  C# _; cMapping from: Abelian Group of order 1 to Set of ideals of K
" W0 T# f. ]3 D2 j; v) TAbelian Group isomorphic to Z/2. Q& H: c4 s6 H: X$ h6 h) X
Defined on 1 generator% t1 B6 ~. ]+ d; S- E$ }1 n, D5 p
Relations:+ O  b: F9 G9 R& t% F
    2*$.1 = 04 N8 `5 Y6 ]' O* G& [
Mapping from: Abelian Group isomorphic to Z/28 F% B2 G3 f: f/ _$ Y& ]
Defined on 1 generator) I- b  Q% \* m1 ?* s# T- j, I8 @2 h& M8 H
Relations:% x+ f4 [( Y3 b) M
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
5 C' I# u0 w* \3 Iinverse]
3 `! k) H2 x  Zfalse3 r: E+ G+ O- u% A; [, }! W+ u5 y
7: @9 k- K) u) V) N: J1 s. R8 k. Z
14: H1 a; d4 {6 P) k3 C8 V. b9 u
28
作者: lilianjie    时间: 2012-1-9 20:44
本帖最后由 lilianjie 于 2012-1-10 11:23 编辑 ; L! I3 m4 g' e+ d/ ?! O

/ F0 a" T2 l! N4 I 11.JPG 5 L( `9 E& E! K% p8 F
& s+ e/ Y7 v2 x( f( H2 ]
3212.JPG . `. g8 D9 b% }( y, ]) n0 b) L  K

& J' I7 v6 d! v0 i7 I 123.JPG
" v# A" }3 R& K; `) a0 e# a! [) t$ p) Q6 \! R# T5 ~
分圆域：
. b* U; U0 z3 m6 c  b, Z3 g$ rC:=CyclotomicField(5);C;
% s* q, Q- _5 c5 x3 {2 i7 U/ DCyclotomicPolynomial(5);1 d9 @: s' j2 T, O" U8 J( @
C:=CyclotomicField(6);C;1 ^" m! I/ l" a& }' a
CyclotomicPolynomial(6);5 S3 j. C5 Q8 U4 D& M
CC:=CyclotomicField(7);CC;
0 [4 x% h+ u1 s. s0 E& LCyclotomicPolynomial(7);( B3 h. r  l, i- {" b% ?
MinimalField(CC!7) ;' T/ v5 }) J5 v1 F3 P
MinimalField(CC!8) ;5 A$ o0 ~2 s$ |( |# O
MinimalField(CC!9) ;+ q: k' u; A7 Y4 O
MinimalCyclotomicField(CC!7) ;% r, M/ o4 d1 m7 B, [
RootOfUnity(11);RootOfUnity(111);
2 X2 S: ^) X6 t) }9 H* rMinimise(CC!123);
- w, ?! j( e! h" }9 }1 R/ XConductor(CC) ;* ^+ B, l2 [. A  l
CyclotomicOrder(CC) ;
- P5 d! ]4 {, |6 z/ ^3 q
2 p( B7 ]; u9 ~/ ?) }CyclotomicAutomorphismGroup(CC) ;% {/ ~% E! L% E% ~' r

( U0 I, R; S8 ZCyclotomic Field of order 5 and degree 4' d4 O1 V: O3 q* [, I: S. u6 D
$.1^4 + $.1^3 + $.1^2 + $.1 + 1+ s; W  j$ {/ F& N
Cyclotomic Field of order 6 and degree 2
' n7 m! `6 h3 `/ Y$.1^2 - $.1 + 10 m' h9 N9 x% O8 v4 H7 g% z
Cyclotomic Field of order 7 and degree 61 }& }/ C- i# o4 D- u
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
( n0 U4 ~% S+ u& c! KRational Field
: C: ~8 `( x& ^* O% wRational Field
" @0 F+ o! `$ q) P  l( [  IRational Field
( P/ g. J9 b/ P( b3 B! z- k7 L# pRational Field
7 x6 w; n! `+ A  J( Uzeta_11
5 t0 V$ `1 a$ e2 q/ B8 Vzeta_111
  Z( |  S- Z2 n3 M123
3 e# U9 }/ a$ \4 g. Y, ^7) d% W5 q0 r% i6 ^6 t
77 b$ t% t: E8 q1 p
Permutation group acting on a set of cardinality 6& m" }% q( z" b. O1 V7 f6 s
Order = 6 = 2 * 3& R4 ~" c3 J9 T1 B0 ^9 [) g
    (1, 2)(3, 5)(4, 6)+ ~& d2 \- h3 V; O# e: u( @/ D
    (1, 3, 6, 2, 5, 4)
8 |: m4 Q/ S+ FMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
( s7 h% F( z* h( }2 [7 A/ zCC% M2 n; P& K2 w8 S
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
- S0 V) o2 e3 p+ W: PDegree 6, Order 2 * 3 and; I( l0 U# d- b6 s/ u& u
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of : c, s7 \  D. n
CC
作者: lilianjie    时间: 2012-1-10 11:34
本帖最后由 lilianjie 于 2012-1-10 17:41 编辑
  C0 l0 O% ^& \3 S" u  ]
lilianjie 发表于 2012-1-9 20:44 ' j  E7 _* j& P; e/ O( ]% H
分圆域：+ t# D( E* F, T
C:=CyclotomicField(5);C;
$ B) q) j6 Y+ Y6 rCyclotomicPolynomial(5);
5 `3 V/ u4 O& M
0 _: R* c, o! c* E2 s" E, N, k
分圆域：! O6 G, H3 d1 O+ k' x
分圆域：123. C& R) T* m' w9 n& k/ x

5 T" B' L% i. x0 U8 {; n- MR.<x> = Q[]7 J, S* V- Z& E! ?* J
F8 = factor(x^8 - 1)$ S% U1 C+ O1 U: C( u8 `
F85 m6 Z; e! L' s# H( ^+ V8 q- e  q0 D
' a0 E. L. I8 v8 ~1 _
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) - K; [, X% q( V5 ~

8 c! y5 R4 R& C! KQ<x> := QuadraticField(8);Q;
1 h5 p( f' k( Z# \: qC:=CyclotomicField(8);C;/ r9 J! B/ L( x$ j
FF:=CyclotomicPolynomial(8);FF;6 m; m* y3 F# h+ g+ w

. F0 Q( F, a2 z- v: q0 A4 ~F := QuadraticField(8);1 q0 V6 f( l# t9 l* Y: E. H, O
F;
+ S* \7 s/ M* b0 a4 X. V  [( l+ OD:=Factorization(FF) ;D;+ p/ h* `3 \$ S$ ]' K( l+ a
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
( p2 v5 ~9 B/ N$ y4 d2 C- Z1 RCyclotomic Field of order 8 and degree 4
0 j" `, E  Q& ?- X6 {2 e$.1^4 + 1
$ Y, ^  E% {" A8 X+ P! b8 X, vQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# [$ n7 g- A) M6 ^% s[
, @7 X& x/ J. K, _  S% C; t2 Z0 r/ o    <$.1^4 + 1, 1>: \( x3 P2 w2 S
]
" R2 n0 }1 N) L3 \$ E6 B5 Y
' `6 E* J( j( q- X4 U/ \2 G' ]R.<x> = QQ[]
8 k) B; A; L  i' z, |1 gF6 = factor(x^6 - 1)
. ~  Z! \# v+ e) mF60 B, [  r6 P+ L; W& \5 ]

+ [7 G; N& u3 t' A(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) ' U6 Q# ?9 G! x  x2 q" n

( `" C" T+ n/ W1 `1 w1 u  G8 t1 bQ<x> := QuadraticField(6);Q;
0 U" W$ R/ e. t$ @% I7 uC:=CyclotomicField(6);C;
0 r4 m$ w3 T" T. a% l! zFF:=CyclotomicPolynomial(6);FF;
# c! l( b9 H: p) J' m3 u: c/ Y; }$ c" C
F := QuadraticField(6);
+ c6 l' |6 _8 @% GF;% O' K6 g& X6 `0 a- ^! d
D:=Factorization(FF) ;D;. t: f! E" ]) f! C- B3 K
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
& F4 F2 F2 y( O7 q2 jCyclotomic Field of order 6 and degree 2
9 L. C$ g7 j8 v' \" @( i4 h$.1^2 - $.1 + 1
+ d* t, C; U  ~8 ?" kQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
  e' c7 c' [* s5 w. e% K% u[2 b4 z: E, J. R7 \1 c- T' v
    <$.1^2 - $.1 + 1, 1>* Q4 e# ]8 d- S0 q+ n1 y
]/ m% [4 @. w* n0 I% F
4 z5 k6 b. q1 C+ X* u, s
R.<x> = QQ[]
2 S' `2 j% x, n; ZF5 = factor(x^10 - 1): F) D7 U! X/ Y: K" r+ V, s( `
F50 U. C3 |0 [- p( u' ^. ?
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
7 C& P: `6 w% [( ~3 N, M9 u1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
1 w6 C& I1 Q+ Q, k2 G) d; u! r5 e2 Y- ~' T' e+ {) F
Q<x> := QuadraticField(10);Q;, h% s, c3 H6 p" a; c0 }: a$ c
C:=CyclotomicField(10);C;
" j- z3 X, G; uFF:=CyclotomicPolynomial(10);FF;4 e; B, P9 L1 U% Q$ |8 o+ c
' l  `- ]8 T: X- j: B
F := QuadraticField(10);
+ H; n+ H" S% r4 X/ {: v6 ~  rF;
$ w' G( o; [7 G5 y3 jD:=Factorization(FF) ;D;! s+ A$ c7 D$ G2 O- i3 g
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
6 C: X. {1 ]; a& \6 D& e+ w( r# ]Cyclotomic Field of order 10 and degree 4
& H: n8 J' y( S5 p7 t6 K: n$.1^4 - $.1^3 + $.1^2 - $.1 + 1' u7 w% c# v/ z3 d/ h- ]
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
6 ^8 K0 i5 S: j6 y3 T[
& h6 ]  L) x7 x    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
: W, v( k* I2 j! v( `7 ?]

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a.JPG

作者: lilianjie1    时间: 2012-1-10 11:55
分圆域：123

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