虚二次域例两（-5/50）

lilianjie

0 N" i: w1 t* T# Y2 G: M8 `
; @( {% F0 {1 K% ?/ w$ K6 EQ5:=QuadraticField(-5) ;
Q5;

& N2 D+ r. m- {6 R( d0 v1 P' v3 ]Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
1 d$ q! j! a9 i" s7 v- U& qM:=MaximalOrder(Q5) ;
M;
* ^/ e: p4 o; H& g+ b: jNumberField(M);
6 q* s, j' N5 N- v# B3 E$ t" 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;
, l" G4 h/ E$ O/ Q9 SIsQuadratic(Q5);
IsQuadratic(S1);
; y( S& S' c7 ]IsQuadratic(S4);
IsQuadratic(S25);
9 x: F% ]- R: S$ WIsQuadratic(S625888888);
, v3 O1 \! B! c2 x( uFactorization(w^2+5);  
0 m  O- I- k. xDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
3 ?: @6 g8 v  KConductor(M);
ClassGroup(Q5) ;
" J5 w( }% _1 U1 @ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
1 r7 X, B$ A0 l8 D, T( _& Z# aPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
4 J" \% z0 R, c0 gNormEquation(Q5, -5) ;
  o$ X4 H* v" A6 U# O4 MNormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
* ~; S2 e  v& ^/ [Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
$ p! l4 A* D5 [/ i& O6 jtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-20
$ _. H% f+ q% ]+ H; _4 X
>> FundamentalUnit(Q5) ;
                  ^
  |+ D! T. ]5 i* W' e) ], `Runtime error in 'FundamentalUnit': Field must have positive discriminant

, L0 |8 B. W' Z. I8 v% q( c7 O2 v
1 p; f9 Y: e1 j! T: ]& o  Z>> FundamentalUnit(M);
) j/ [4 t3 y/ J% P* K1 m                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20
$ ]( z' L1 ?% L* w- D4 o! D
>> Name(M, -5);
       ^
  g, w6 ^* J. B* u2 h" ERuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
0 }0 \% {5 j0 J- [0 A, O
1
8 x% h' e0 g1 ?( K8 sAbelian Group isomorphic to Z/2
Defined on 1 generator
& h3 X( \- `+ Y7 j8 IRelations:
  I0 g7 J. l4 Q. t- ?& x3 d( G    2*$.1 = 0
) G1 h  `- }" ?: F5 kMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
# N) a3 L1 E0 t+ k: a- NRelations:
0 j) S( m9 ?6 |0 `# z4 J  \    2*$.1 = 0 to Set of ideals of M
% l! a; v0 E% |% e( F$ NAbelian Group isomorphic to Z/2
5 f8 S9 s" k" f* v3 c# ?, _Defined on 1 generator
5 z  E# v% b0 uRelations:
  {+ w2 @+ A# T: ?    2*$.1 = 0
  e; f: y: k! p% t9 Z- @! fMapping from: Abelian Group isomorphic to Z/2
$ I2 r/ R# r, }Defined on 1 generator
0 O  J" l) U; j' B1 ~7 LRelations:
    2*$.1 = 0 to Set of ideals of M
9 o  i! h) H  _: Q2 R, A2
2
+ F- c; P9 j1 c- b# XAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
  h7 O( n! `. A1 WRelations:
9 H0 V. u3 I8 V5 E1 {5 R; b6 l    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
/ B  I+ j! n( W* Z' P2
+ x4 k7 \0 [. ^, ZAbelian Group isomorphic to Z/2
- G4 [8 j' ^' Z- S; }- E$ FDefined on 1 generator
Relations:
* n: E+ Q! X/ C& R    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
/ G( k+ k9 ]  E' S* y8 wDefined on 1 generator
Relations:
2 J% G$ s, L5 i! B    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
5 W1 a; M9 T0 N: l! wAbelian Group isomorphic to Z/2
- F/ D" c, ~3 X0 }9 u6 l2 vDefined on 1 generator
  T( g6 [' ?+ V; S9 nRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
8 n" `0 y9 A0 a  D- R/ {( DRelations:
; d7 K0 p1 t; p/ ]( _, c' v    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
: m8 z+ v( e: H& g  \false
$ u4 L- M, I2 f9 Z( @8 v9 ifalse
% z( }) O! `6 g==============


# l. Q* W7 B- \8 J4 e. p$ kQ5:=QuadraticField(-50) ;
; T( G4 q! ?" U- e6 q' lQ5;
5 E1 F' d5 y# j1 ]: h7 Y: P
" i. ~3 v+ F. v' UQ<w> :=PolynomialRing(Q5);Q;
- [# d: X! M' t) {EquationOrder(Q5);
/ r5 O: X" ]4 o# x9 |M:=MaximalOrder(Q5) ;
7 l6 e6 j! l: }, z- AM;
NumberField(M);
. H* i$ ^- O% D% ^6 xS1:=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;
4 W/ `- m8 I9 x# w/ O/ WIsQuadratic(Q5);
5 H- S* @  G! o: S& H% T* PIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
" i5 R( Y1 J( O1 V2 J) B* }2 OIsQuadratic(S625888888);
, w/ q; f* p$ U9 Q& }' H) IFactorization(w^2+50);  
Discriminant(Q5) ;
- ?4 W! X# w( l( ^; e# aFundamentalUnit(Q5) ;
% p* \$ q1 c+ D$ \3 ^2 w1 sFundamentalUnit(M);
! W3 `, s1 j* l: H. C7 E6 m7 o$ n3 rConductor(Q5) ;
* J" N# C6 H. e( Y
$ R: R) }& W( z- K$ P/ x7 D' AName(M, -50);
Conductor(M);
( b) ]* E; n" _! n4 y5 S: TClassGroup(Q5) ;
ClassGroup(M);
6 a' `6 }5 k2 N" s  iClassNumber(Q5) ;
8 j$ M3 g$ l& r6 n! g0 o% aClassNumber(M) ;
9 r+ T" g% U2 I3 dPicardGroup(M) ;
7 n. Y2 m- Q9 U: VPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
3 S) w9 d1 a% m9 ]4 ONormEquation(Q5, -50) ;
& z6 y6 o9 P. V/ r; g* ~8 g" ENormEquation(M, -50) ;

: r% v, f* t4 d: Q- i8 P2 r+ U  vQuadratic 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 $.1^2 + 2 over the Rational Field
# F$ S# M% [8 o! DOrder of conductor 625888888 in Q5
+ H0 M! V; q! Y" U5 L3 K4 itrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
& @  r* \. m1 i4 ~- M& r, S    <w - 5*Q5.1, 1>,
% T$ H/ x* C& {* h% R    <w + 5*Q5.1, 1>
]
-8
" W! ^2 Q6 h/ h- H
; W' L% \$ n0 l+ m>> FundamentalUnit(Q5) ;
                  ^
3 \) q! ~) A$ b: G9 }Runtime error in 'FundamentalUnit': Field must have positive discriminant
% i% E0 r" t# Z6 l( N* |' v* |: ^
. u5 k) e1 E3 p* j& j& i8 {
>> FundamentalUnit(M);
4 ^: v! c3 r% R$ G                  ^
  G/ f9 m% `2 J0 lRuntime error in 'FundamentalUnit': Field must have positive discriminant
# Y6 }" }* M9 _, l: e0 f
1 @) b- }( I- Y8
2 \2 |$ l$ r; {
>> Name(M, -50);
       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
) k6 \9 m/ t) |8 W6 {; Z
1
3 z+ R0 I$ P& N3 t8 T7 M* Z5 u8 ?Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
) k9 d* R) J8 L7 w/ lMapping from: Abelian Group of order 1 to Set of ideals of M
0 Y5 g* s: U$ t7 t  k% d+ y+ ~1
1
* Q( o5 h& B$ t2 f1 K, A' x( xAbelian Group of order 1
- x" T; q% d/ i2 ^' m; AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
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]
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]
false
. ^/ a( v* E/ f; }5 zfalse
: l' _" g4 T' G( l* r

lilianjie

看看-1.-3的两种：
: B; u0 q7 Z/ m% i8 X
Q5:=QuadraticField(-1) ;
+ x& V7 e7 }, t2 M2 }Q5;
5 v( Y5 w( S$ y9 N# `, j3 ~- h
Q<w> :=PolynomialRing(Q5);Q;
7 L2 d) l& z6 _1 e- KEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
8 H! r, G+ `9 O5 t$ c+ P6 C4 O( {M;
5 @9 n* [! ~; qNumberField(M);
2 a( p8 W  }( ]5 |' y3 Y' LS1:=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);
7 y: Q, D" }% W7 z& c0 TIsQuadratic(S1);
  w2 D' w1 L6 ^+ T; DIsQuadratic(S4);
IsQuadratic(S25);
: A% Y% |& v" @5 dIsQuadratic(S625888888);
) a2 c) ]& l! p# L; I! \3 WFactorization(w^2+1);  
7 t( _' [2 y6 z- a% s- S- O& M. }; qDiscriminant(Q5) ;
. C( y/ ]2 z4 j6 R  n3 E: W5 d  Y& zFundamentalUnit(Q5) ;
FundamentalUnit(M);
  F7 t) h# Z5 IConductor(Q5) ;

( Z5 w3 Y5 P8 Y3 m8 j0 yName(M, -1);
4 o" j3 @9 V( UConductor(M);
ClassGroup(Q5) ;
' [' [& j+ i" L" TClassGroup(M);
ClassNumber(Q5) ;
- d$ D" }0 }# s: v- ?; jClassNumber(M) ;
PicardGroup(M) ;
$ k! Q6 H; K) KPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
5 j$ V8 Q# ~0 y, i7 R, d/ {% }# M$ INormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
3 E/ g4 M1 t1 G5 b$ _Univariate Polynomial Ring in w over Q5
& t% q( |6 p# u: q  V1 BEquation Order of conductor 1 in Q5
$ w9 r3 r9 {) H9 F4 vMaximal Equation Order of Q5
& I. H- {- L, A! F# `  @) B9 rQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
' l' I, a4 E* Btrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
* N3 m) u: q# x3 C' y$ Ztrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
: k% p. k2 J" E* T! Z7 c  c[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
3 R' l& Z6 V) ^9 t; a- g$ e+ _  d  D-4

>> FundamentalUnit(Q5) ;
# B( n  @( z8 M3 z0 a  s3 Q                  ^
, v- w4 ?. l" v9 S8 _7 MRuntime error in 'FundamentalUnit': Field must have positive discriminant
" i. z# J0 n3 g' M& `$ P% I& [

>> FundamentalUnit(M);
; v0 y2 U! g" [$ ?, `6 U& U* t                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

7 K1 R2 u5 D, P0 K- U4
, ^5 e1 I: P! N% y
( v" B  C. d+ W! P>> Name(M, -1);
       ^
) H! u0 `! t* R1 F8 \" J# c3 nRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
/ `# S7 ^0 O7 Q/ t* G( \7 |, SAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
; `8 Z0 Z6 p  u" H1 b' p1
Abelian Group of order 1
3 }1 d7 K; X- n- u4 k/ W/ ZMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
$ t6 m4 y" r% j4 V" pinverse]
( ?+ t( P9 r9 M0 a0 b, R. n1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% ~1 h1 j! m9 z! G8 s4 T-4 given by a rule [no inverse]
( G" f8 O2 ?* n: [' LAbelian Group of order 1
3 o" X! f7 p  S/ I8 X! T" IMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
! \8 l9 ]9 Z) @1 Lfalse
  h! _3 X, j# b/ S! |  @2 c===============

Q5:=QuadraticField(-3) ;
! n. n7 H) E/ h* c  s. H" \- bQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
) ]" L- ?/ r0 f5 Z& p6 A* rM:=MaximalOrder(Q5) ;
' n" ^: j. y, Q- h& |& q& ^M;
NumberField(M);
) R; \, f' ]8 R# D8 m/ V1 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;
* S2 b; ?9 b# H9 ]$ z/ C: R, pIsQuadratic(Q5);
IsQuadratic(S1);
" J: X) j% ]2 M5 ?2 WIsQuadratic(S4);
2 _+ ~5 o1 q; C1 T# RIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+3);  
4 f% j1 ~. Z; ]& m  T. w1 yDiscriminant(Q5) ;
+ F# l! E* Y0 D) g4 Q* xFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
* ^& s( `7 L/ e* ?- d6 w5 F; u+ ^
Name(M, -3);
0 F0 ]( Y# c( @  Z5 w* O4 kConductor(M);
! b! T- Z. Y# K1 `2 s$ RClassGroup(Q5) ;
8 D8 E# o! F/ M% o0 LClassGroup(M);
ClassNumber(Q5) ;
, z" t  {! z+ u& D  NClassNumber(M) ;
; `7 Q" V  `5 G5 gPicardGroup(M) ;
3 Q1 s7 V& p# D5 G1 M  wPicardNumber(M) ;
/ V* j4 T0 ?- g2 [: O* f
5 w. r! g9 }3 o) {+ g( D. f5 k2 ^( ZQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;

! a. C7 Q9 n6 [; O5 q; ]' n: \& e$ gQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
, ~! k6 h1 b5 a+ L/ M. _Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
& `6 b- n0 M: F/ I' e+ b4 @& jOrder of conductor 625888888 in Q5
) M) s/ K+ X% _4 ztrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
! R) F- n- V" {- ], z% h3 s0 Gtrue Maximal Order of Q5
: ]" R& E  {  L  M! f, ?true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
' b7 v: H) r: ^6 {[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
) @) S7 s1 x" J; ?-3
" Q  q( V* m  y2 x
>> FundamentalUnit(Q5) ;
' \8 R. A1 D6 I1 m: T# {! }3 H                  ^
/ G9 o& D( Q7 ERuntime error in 'FundamentalUnit': Field must have positive discriminant


& ]- @: f0 \" u6 ~6 Q7 m>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3
% h. n4 |) i- S/ ?) v+ V3 c
>> Name(M, -3);
$ N+ w% P: X8 P, [6 i+ q       ^
' x, a6 M6 U1 iRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
7 \! i; \1 M4 [1 I2 E$ Q' D
6 h5 R1 i- d8 m! @7 X9 i$ X1
: T8 V+ y/ Z8 Y! k, cAbelian Group of order 1
3 E$ `. @4 q) ]) I- w4 k0 yMapping from: Abelian Group of order 1 to Set of ideals of M
6 s  |. x# E" D" c% Y9 _( PAbelian Group of order 1
8 |3 `1 v; M5 P5 d& V3 cMapping from: Abelian Group of order 1 to Set of ideals of M
0 h7 H/ ~* Q: B. t" B" _1
1
) T! [+ }, X4 IAbelian Group of order 1
5 m& u$ ^8 z, ^7 J" l  hMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
3 J1 d) d/ p. i- s* p' S* q0 O8 @inverse]
' m7 x3 q5 n* {+ I- k) g( w/ Y1
Abelian Group of order 1
  y3 ^# x7 ]% uMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
1 @; q+ \2 [# W1 H/ B% Q9 P1 H3 a5 h. ~-3 given by a rule [no inverse]
1 V! _9 p( E2 L) s' Gfalse
false

孤寂冷逍遥

lilianjie

  P. B; Y3 ^5 _Dirichlet character
! T. i' t5 n. ]; PDirichlet class number formula
& r* R. X. E" r3 |3 ~
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
% A: J! T) ]+ d
-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

, n/ u0 _3 [( t; W" O-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
h=-6/（2*3）*Σ[1*1+（2*（-1）]=1

-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，
0 [1 A' c2 K4 |* {3 E. J
( D' B/ d; w4 J8 Y. R  }3 @, c
' \# O6 j7 u# U4 n9 A: u- W$ Q( l5 g
3 v5 t& N" M6 {6 ih=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

5 R- Q; \+ R3 ]- x7 i
# k6 v: Z7 W3 Q. a; k9 y4 n
-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

) y+ x0 i8 r' {( X0 @F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
' k  P% D; [7 S% A# }7 h/ bK:=MaximalOrder(KK);
( w4 E( S8 F7 vConductor(KK);
. n' l: V- R" Z" D' N2 \% _6 zClassGroup(KK) ;
- E  F$ g$ L5 a$ h  Y! _& O% CQuadraticClassGroupTwoPart(KK) ;
4 j' d# N/ ]1 C7 T, L% i; fNormEquation(F, 7);
5 K5 b0 R" q7 e" y0 o2 N6 p$ RA:=K!7;A;
B:=K!14;B;
Discriminant(KK)

2 }2 R# M) ?& BQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
* a  m' n* q/ a" V! i( @6 E5 DAbelian Group of order 1
  Q& p3 x& b. {Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
$ W2 j1 R" M$ I9 z$ a/ n/ iDefined on 1 generator
Relations:
  t+ U: ~2 V9 f    2*$.1 = 0
" [+ S) X) d8 i1 wMapping from: Abelian Group isomorphic to Z/2
' [+ d9 m% v2 z  s6 Q' G2 I' s6 C) gDefined on 1 generator
' y" {+ Z/ o7 X& o4 \4 J- v; xRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
* N. _# u# `3 b14
28

lilianjie

0 {0 r  H, P: n5 T3 F

% T# g! B2 o; U* Y9 f* g- }8 a6 G9 D7 Y
& o1 \6 z: u% Q2 x7 L/ P$ Z* u  y
! M( z3 Z0 f  y- N# I& t

$ l& \$ C8 x+ }- a# A# j1 Y
+ Q; c3 }. m  b% B分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
2 M$ W% F2 W% l. KC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
$ A1 S6 y+ H: a# m$ b0 x. BCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
; L& j4 m! D( P$ x5 oMinimalField(CC!8) ;
! r7 d  d8 ~  ^MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
: _9 _) V6 I% [RootOfUnity(11);RootOfUnity(111);
4 ?2 t, O; A& c, [Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

: s+ u+ p0 r$ c/ G# cCyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
% q3 s, v9 v8 K; a: y$.1^4 + $.1^3 + $.1^2 + $.1 + 1
/ X# D  t/ J# r# @" f) `& j9 [6 ECyclotomic Field of order 6 and degree 2
3 }1 ?$ f8 M, a$.1^2 - $.1 + 1
6 N7 m4 t: H$ @  D5 X0 FCyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
. f: E. b6 B+ H! gRational Field
Rational Field
Rational Field
7 k7 {& X% A' C1 }+ nRational Field
zeta_11
: m& T' M: k! D7 V7 L0 H2 mzeta_111
3 q- i) p* S6 S6 c, q" M123
- L, D5 a, y' F1 K) C7
7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
. p# N4 ]/ _: f" ]7 K9 D2 DMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
8 [/ e0 q2 ?: v) H6 b6 h  Y0 fComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
( |' k5 i$ O9 F9 {" R9 jMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
5 g6 w4 a9 H- m) I7 i* ECC

lilianjie

" g8 f7 k+ q5 q

lilianjie 发表于 2012-1-9 20:44
分圆域：
/ v8 R4 A& D4 r0 h4 FC:=CyclotomicField(5);C;
9 `" U. `  n* V+ ?! ?# p- _CyclotomicPolynomial(5);

# m8 Z  |6 d5 W& F  q  e% R分圆域：
/ g  g5 U9 }6 w- ^分圆域：123

R.<x> = Q[]
& g( O' c' G- \F8 = factor(x^8 - 1)
1 l# H, \+ l% ?" vF8

  V- G- v: e1 f2 ]8 T! h! F' W(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
, e6 P- M! r! f. D
Q<x> := QuadraticField(8);Q;
2 s  z1 M. {2 Q/ q/ g5 S9 pC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

& m/ x1 c4 f# v! v8 j# O4 ?! MF := QuadraticField(8);
F;
/ D# R1 \! w6 j# h0 e/ k. TD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
4 r5 b" \4 ?- |+ H: \2 y    <$.1^4 + 1, 1>
]
; _, N$ R  k  T! W! t; S6 A
R.<x> = QQ[]
) T' }8 ~+ G) I+ W0 hF6 = factor(x^6 - 1)
F6
- ?7 n& _  W# U5 W
4 D& Q. j8 w' ]" Z1 e4 R(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
% Y6 t6 O, @. `
Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
2 |  O* E9 N- d  UFF:=CyclotomicPolynomial(6);FF;
! T( k- l' M( d
F := QuadraticField(6);
# A. |; S( S& X* V% UF;
% K; }2 r( P* K1 p8 O$ E+ TD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
' \" U, ?- g5 ~Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
* x1 q+ f6 K5 c9 Y! i+ kQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
" D& u8 M" ~- a: N9 w* Z- G[
    <$.1^2 - $.1 + 1, 1>
) U4 t" c! c( w( `5 s% M]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
4 Z. p$ S0 p- ^" h1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

  d0 _+ U. o6 j+ o, fQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
; a. ~" W5 U: z! g$ B4 DF;
0 Y- @* G& K/ U( h- ED:=Factorization(FF) ;D;
1 i1 y, I) L( @3 D# g7 IQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
5 L( C: I, Q7 k% }) c/ S$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
' q- I. }0 r2 }) G[
. [8 k' O' l* C1 g: h    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123