虚二次域例两（-5/50）

lilianjie

3 I' Q6 v4 g4 k4 r$ I( p9 I4 A
" A7 l/ Z+ N, z- O$ X' N5 `Q5:=QuadraticField(-5) ;
! ?) d: \4 J- UQ5;
- Q- O0 b: ~0 e5 `" Z5 z1 |, t
0 _) x' l+ X$ V( ]. @! RQ<w> :=PolynomialRing(Q5);Q;
! J1 w. x% B5 t+ J( F1 S2 _EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
. [, }+ Q$ @+ u% XM;
NumberField(M);
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;
IsQuadratic(Q5);
IsQuadratic(S1);
4 x5 D* b4 `+ C+ L/ |+ @IsQuadratic(S4);
IsQuadratic(S25);
9 Z5 i' y% e+ [2 AIsQuadratic(S625888888);
. X# Q5 {9 P- @+ u4 X) a7 pFactorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
" Y0 W6 j) ~- f$ G( [" ]
/ M4 I, J1 P/ v! |3 T) Z) p# eName(M, -5);
Conductor(M);
( k/ d$ k  u: ~, h. ^ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
0 n8 t( J. |% |! h! S1 {- V7 n; CClassNumber(M) ;
- ~! j" j/ I) X1 g$ PPicardGroup(M) ;
+ p; d4 ?1 t1 ]* g9 X6 QPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
1 z- [; z1 G: `. ?; P, g  w2 OQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
+ ^, |' P% Q$ s" K+ ^Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
0 N! x7 v, Z6 fQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
' X6 y$ {; v( j0 x  p' b$ FOrder of conductor 625888888 in Q5
+ V% {% }; N% T$ o) M* P: `true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
6 ~) ]- ^3 H/ N5 Utrue Maximal Equation Order of Q5
  Y  _& t4 D; S! d, [7 x( [' T4 R& Ztrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
- B& K# {$ U+ C9 K. I2 V    <w + Q5.1, 1>
]
; ?7 O( A) K& s) Q8 _/ B' M-20
+ j+ V! H1 m! P" J+ n6 r
>> FundamentalUnit(Q5) ;
6 F" O, t9 ]2 Z9 Q                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
$ y; Z$ I* r3 v9 E7 a                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

& ~& G4 j! T* m: ^9 x" _20

8 V7 e! p$ G: D' y  j1 z  `>> Name(M, -5);
+ p0 S. [) j* A/ ]4 z/ m       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
Abelian Group isomorphic to Z/2
% r! r& A8 f) U+ h* J& G2 SDefined on 1 generator
3 I' ]; @1 ~' w. R; \Relations:
% O5 A/ `* u% d& P0 X! Y9 X2 h. x/ z    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
0 W7 ^+ v$ A, @' G+ W6 X+ x; _Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
) q2 j( U9 R3 r& X8 }" MRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
9 i/ `3 l6 b6 s8 f: WDefined on 1 generator
8 T5 r) O- O2 ^5 X0 O4 o; `Relations:
    2*$.1 = 0 to Set of ideals of M
+ X! k5 Z% ~. o7 c) J2 c2
& [4 ?! h- O. f8 I5 Z2
Abelian Group isomorphic to Z/2
- F# M& j; F. e' `8 Q9 H- pDefined on 1 generator
2 B& A% w1 I. N  p2 D) |, @6 wRelations:
, B( B) z  P& W& w    2*$.1 = 0
' z6 n& }5 x: r) k9 `) tMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
) h1 }4 M* x& r- V" }' c2
Abelian Group isomorphic to Z/2
Defined on 1 generator
7 L0 e; H$ F; ^: [2 J4 }* RRelations:
4 {2 G. c, T: l  h/ Y+ F    2*$.1 = 0
& x# c7 G, i5 ^Mapping from: Abelian Group isomorphic to Z/2
% B/ i+ X; C- ^Defined on 1 generator
- v7 ]. q2 J1 K) QRelations:
9 E* X: ]* q/ v- X    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
9 T+ ?2 M; O; e! N; J5 xinverse]
" z, E" w' F% S( [Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
9 Y& B. s( y" L- L3 v    2*$.1 = 0
+ j6 r( s; K! a4 u6 r! T- oMapping from: Abelian Group isomorphic to Z/2
9 q. h% z  A; o1 M( BDefined on 1 generator
# @5 [+ A3 q2 f. x& B2 P& @5 `/ kRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
- t4 d& d1 r- r/ dfalse
: q/ N- I+ l- i) Efalse
==============

+ n0 B( f$ e8 D2 y2 B' Y
Q5:=QuadraticField(-50) ;
3 X* O% s; r0 gQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
7 x4 P6 _# }3 H% S  }$ zM:=MaximalOrder(Q5) ;
7 G  Q  k5 I: l  u) |$ w% {# SM;
NumberField(M);
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;
4 Q$ @  w0 T2 |( RIsQuadratic(Q5);
( r/ ^5 A; j# W+ nIsQuadratic(S1);
IsQuadratic(S4);
2 _% y3 i; |; y2 bIsQuadratic(S25);
- d" |  Q3 k7 @4 q% J; SIsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
5 z( {( C7 O1 o6 p2 uFundamentalUnit(Q5) ;
FundamentalUnit(M);
  }- T4 r& w1 q$ R7 a* TConductor(Q5) ;
1 z0 M" Z0 [* r0 z+ C
" T' W! n4 X% [0 Q" SName(M, -50);
Conductor(M);
: T" D2 X6 o" x' sClassGroup(Q5) ;
ClassGroup(M);
# ~' K6 \* J. W+ r# Q  s6 mClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
# H4 \# m# v# Y/ i% z3 S+ y8 r
; g. |+ A7 m, h( aQuadraticClassGroupTwoPart(Q5);
8 D% t: B- [+ H" m6 j: v$ _QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
3 w, z, x) ]9 L$ U3 V, F8 _! s- iNormEquation(M, -50) ;
- M* m- D- b8 p
0 Q% s0 \1 S- f5 _, Z% _7 nQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
# g/ _+ `# i, y- m$ XUnivariate Polynomial Ring in w over Q5
2 c& j: M8 z5 \6 x, N( vEquation Order of conductor 1 in Q5
( x0 P5 `6 a- s+ @7 `+ f$ }9 VMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
% G# c4 O* i/ D  S/ Q$ z+ }( U: DOrder of conductor 625888888 in Q5
3 Z7 R$ c. g' ytrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
  P, e+ d" g% O% e& o, b4 Q! w+ Wtrue Maximal Equation Order of Q5
0 E4 j9 D; O1 wtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
& T3 n0 V( \9 j( Q% [true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
8 c4 b, \+ L! }8 E, T5 B( Y]
3 u. U4 E. H3 E/ L4 U- l-8

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
- ^$ e/ A$ V9 v, r* G7 Y& p+ A7 B) l
- C9 P8 p1 j7 F) Z
  r) l; p: c3 T$ i>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
7 `0 I1 W7 S% o
2 I' ]  @' T% `$ i4 W( e8

>> Name(M, -50);
1 z1 b7 X' |7 p5 J       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
! n6 v: Q, N  C& o6 V, s$ h
1
3 Z8 E/ T+ I5 j5 L  |' G% ?. @Abelian Group of order 1
. H4 C7 g/ K! r2 s7 r: r$ D2 F, bMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
' w. q) C! i- e# E3 r, S# u$ GAbelian Group of order 1
/ ]# v/ z1 n* |) W0 O/ NMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
, C3 X! \0 o5 P! L7 pinverse]
1
& O/ h( \$ l, S+ z' D( }: |Abelian Group of order 1
( f+ z, G: @" G8 K' j% l& w1 bMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
# s6 e  m7 F  Y8 C; i  V-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]
9 S8 N3 r1 \; }: F. bfalse
* P: |2 }- X5 e! C2 `& f) ^false

lilianjie

看看-1.-3的两种：

8 \" p, j& U7 C' ~3 Q; z( LQ5:=QuadraticField(-1) ;
! T% ~. t! s: R) `* yQ5;

Q<w> :=PolynomialRing(Q5);Q;
2 e: ~. P* {1 z& y: dEquationOrder(Q5);
/ ]- O5 g, e# ~M:=MaximalOrder(Q5) ;
M;
NumberField(M);
" A, Y3 G- w1 a' p8 nS1:=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;
3 f% }2 p* v( {IsQuadratic(Q5);
IsQuadratic(S1);
) o$ m- I) Z, e! QIsQuadratic(S4);
4 O$ `7 V1 x: Y/ PIsQuadratic(S25);
IsQuadratic(S625888888);
$ q: A2 ?- j& C4 K7 b6 W1 oFactorization(w^2+1);  
, m5 K: B! ~+ L, u1 \( `4 {Discriminant(Q5) ;
  y* D2 _: x+ d: EFundamentalUnit(Q5) ;
! R7 v4 H3 f6 Z2 O: {, k7 KFundamentalUnit(M);
5 K- v8 h# M7 \# o- @- VConductor(Q5) ;
: _9 ]2 I1 k1 s* e
, {. s" u5 D( p& \Name(M, -1);
  Q! p, e- d* R# G/ C% |0 `Conductor(M);
4 Q- h1 R# I+ ^4 C3 _ClassGroup(Q5) ;
ClassGroup(M);
7 v; H) J: t0 K% t% S5 \  cClassNumber(Q5) ;
( E, g- [' N- O7 o- @ClassNumber(M) ;
PicardGroup(M) ;
& c. j2 p) ^  L  }, [PicardNumber(M) ;

  n4 p# t. p% r5 bQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
) V" I% K5 x# ~' ?; Y# BNormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
7 I, _# _9 g1 J" |% `1 q8 ^+ t1 pUnivariate Polynomial Ring in w over Q5
7 o% q2 J  n% G1 t8 JEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
; O' f- D3 i2 M+ lOrder of conductor 625888888 in Q5
+ d) ~- j- j3 `2 M- Qtrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
% N; i4 W6 h; ~  e" v4 ltrue Maximal Equation Order of Q5
8 l+ T$ U, S4 jtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
1 t; N/ `; _/ y( T& @  Ktrue Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
+ x# a0 V+ r  d- F    <w + Q5.1, 1>
]
9 S6 M. u  L& O+ D7 Y3 I2 p8 @-4

>> FundamentalUnit(Q5) ;
                  ^
' r! d' g4 g: T5 s- _' j3 J" iRuntime error in 'FundamentalUnit': Field must have positive discriminant
8 G; e, H# L, `4 j' E2 P7 W- j

>> FundamentalUnit(M);
                  ^
+ o) I! F" c- {  `1 E5 bRuntime error in 'FundamentalUnit': Field must have positive discriminant

4

>> Name(M, -1);
) K% d: o5 n4 B/ I& _% _( u5 O' a       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

) ^) B3 f4 }- z, Z' q& S1
6 P" G2 Q4 _1 |5 `6 U0 Q- P1 Z" ?: fAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
  p+ ^/ ?- z9 C# l% wMapping from: Abelian Group of order 1 to Set of ideals of M
1
+ r) X5 r/ U. Q& U5 ~1
3 E% t8 ^+ n! _- V1 T( A6 t7 yAbelian 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
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. g1 c" o% x. Z1 Z4 V-4 given by a rule [no inverse]
; X& F9 i1 h) V0 VAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
false
===============
: I9 ]0 d# l7 J$ ~
# B: p, k  S5 [; fQ5:=QuadraticField(-3) ;
/ Z; j" H) U7 o5 [Q5;
+ C' o+ Y& v1 S4 U2 M6 e
Q<w> :=PolynomialRing(Q5);Q;
1 o# V# p9 B* U2 P! ^% i, ~EquationOrder(Q5);
* P/ T0 e8 ]. C0 x* c6 C5 zM:=MaximalOrder(Q5) ;
9 i" B; A  T& I6 k# V7 r. Z9 M; tM;
0 t- [& G% h4 ~% FNumberField(M);
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;
IsQuadratic(Q5);
. d, A9 U3 k2 ?IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
3 \+ K3 z$ [3 D: ?0 g: T6 IFactorization(w^2+3);  
Discriminant(Q5) ;
' x0 a/ ?  v# \& O3 T$ j- p8 k+ gFundamentalUnit(Q5) ;
% @0 w0 B1 l* L( vFundamentalUnit(M);
6 q: H* O, t) @+ |Conductor(Q5) ;

/ Q1 J) Q. S& g. m$ @7 v% a: WName(M, -3);
Conductor(M);
4 p( `6 ]* v+ m& @ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
& t$ a+ m2 I. X: `/ z. wClassNumber(M) ;
9 r9 Y- r! |6 M8 ^2 ?" D& PPicardGroup(M) ;
9 B, h$ \6 R% ?, e" r! P+ p) GPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  S4 a4 ?2 z# A! q7 xNormEquation(Q5, -3) ;
2 y1 y; H. j' q+ h7 eNormEquation(M, -3) ;

5 A. T+ l: F; a. A+ X4 e* c# i% X+ `Quadratic 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
% S0 Q" Q, U" N" ~4 {# ^; oMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
" o+ x0 c' d' V. Itrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
4 v9 `5 m6 k8 Ntrue Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
5 G' V. L* \. C1 S$ P5 Itrue Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
% y; g1 F+ z2 b0 p]
-3

1 ^0 v4 f3 g% R; e( N>> FundamentalUnit(Q5) ;
! ^8 j4 B8 y6 w( P8 k/ k" y                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
: F/ _, N8 m  Q# Y
( O- l( ^% t% U' l3 ^$ A3

( E0 {. \2 X4 X! K>> Name(M, -3);
8 @6 t& c) F: |7 H9 V# Y       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
, W* ]8 q- z7 {3 {% f% ?
1
4 h9 o3 n8 f- Q% m  M5 M; i7 GAbelian Group of order 1
; k0 t" c) a+ _' r' B8 kMapping from: Abelian Group of order 1 to Set of ideals of M
: I: ?3 X- X1 ~4 _% gAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
* W# d" M& _1 I3 s5 v1
1
Abelian Group of order 1
% u0 Z! G! B6 l5 F1 u) w! I. Q3 yMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1 b$ ?& {" W+ R/ B, O1
$ K! @4 ?+ t3 J0 w8 Z3 d2 JAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
* `& B5 a4 {- o' d, t. Y2 H3 FAbelian Group of order 1
& g' P$ t, ~. _8 h5 gMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- t5 h* V0 H" l9 j-3 given by a rule [no inverse]
  P8 J8 y- j% Q5 c, Cfalse
- E* d' ^7 v9 T- r1 `) o6 q8 @1 bfalse

孤寂冷逍遥

lilianjie

* c% g* G- l, c8 v' l9 x$ a" p5 J& g
& t5 x/ ]. k& e! G1 l4 pDirichlet character
$ ^; N. h, y" F5 E3 R& v1 LDirichlet class number formula

8 B' a$ @& Y. ]0 N+ O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
( ~$ y* O" O' \. o$ p- Q( h: g& G
-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
: ~: ^/ @6 j' v% V! m! I
. h/ D5 _9 Z5 N+ m/ n% l-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
4 ~7 F7 y7 r. n' ?8 d  b4 A: D! {h=-6/（2*3）*Σ[1*1+（2*（-1）]=1

+ U( Q( K5 j* Y2 F+ C' ]) n3 c-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，
( }  Z( N+ v9 Z4 X' L/ L

3 O+ O9 K) l& f) b. T
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

5 ], g& U/ q& f# E& k6 U

  h  R" v, V9 y4 w9 j5 ~-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

- `" G, r2 V6 \' x+ o5 b
! `6 }- q. B9 ^  j  [+ z& tF := QuadraticField(NextPrime(5));

4 A. g0 Q0 s8 X& A0 i/ SKK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
ClassGroup(KK) ;
( [9 F; E  Z6 k5 IQuadraticClassGroupTwoPart(KK) ;
0 l" h; K% C  F! T+ eNormEquation(F, 7);
0 d1 m- |* x2 _; l6 D( T! F  KA:=K!7;A;
B:=K!14;B;
Discriminant(KK)
8 k5 D0 j) B' u# F0 z) Z  \
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
7 v% b  ]9 n  o( \2 UMapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
# N  W& l) Z4 C1 O/ }) [# tRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
0 Z) s1 p1 h. o9 V5 N; lRelations:
/ ^+ K3 L& D. O2 y1 w    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
14
' ]* j* |4 L: a' K28

lilianjie

0 U) w% T2 U" F; A  R) B+ ]6 F

- r0 d/ _& R9 [+ l7 M3 N$ @- _
0 V) U/ D! \6 F6 h, o. y2 y

0 D/ D% z+ S: i2 _+ h3 X! n

5 x0 L. ~  Y' ]+ U+ [+ Y1 E0 p# u分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
2 r6 n' B0 K( A' I* T, f3 r  A0 dMinimalField(CC!7) ;
MinimalField(CC!8) ;
5 w! h- a* U/ g- J; ]/ i/ RMinimalField(CC!9) ;
4 \& j, `1 V$ V* B; `MinimalCyclotomicField(CC!7) ;
/ [2 A" W2 h6 iRootOfUnity(11);RootOfUnity(111);
4 ]# j' u/ s* DMinimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

0 q5 r% a4 K3 v  f4 c! F3 D. _- [CyclotomicAutomorphismGroup(CC) ;
* R5 R+ F4 I9 }; V/ l; N
Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
2 R# u' j6 V0 u5 K+ h6 nCyclotomic Field of order 6 and degree 2
; V9 g! E# B7 w. R$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
( u! P8 \# D, m* K; s$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
& ?2 ~. y9 P$ }/ w' m5 l6 W$ mRational Field
Rational Field
% i, N  D+ I' Y7 t  CRational Field
) f# w6 w" L, I$ X- czeta_11
zeta_111
123
$ Q+ g$ E! q  U+ x& K# q* J7
( @. G/ j5 N# z5 R/ ?" e7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
0 ?' z' s/ g% n" D    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
( r; K. }* g; S& q. YMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
9 Y! O/ {6 F# V$ t  a6 \1 NCC
2 b! ^. _5 i+ f, |1 c0 bComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
, O4 H% W) u  t& X: o* a3 vMapping 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：
, s' K; T2 S  R2 h! D% d' f7 dC:=CyclotomicField(5);C;
1 y. ~9 t+ M9 n# Y: D! @3 ?, Y) P$ UCyclotomicPolynomial(5);

3 l& Y3 P/ i4 S分圆域：
6 D/ [. Y# `0 G6 \3 u* I+ r分圆域：123
  w0 O8 |, N* A  u* u
R.<x> = Q[]
F8 = factor(x^8 - 1)
7 q- r7 K# ^! G" H8 @- GF8

' z& n& J% v4 t2 A. K5 M+ m4 v(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

$ U' b' u0 R9 v- qQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

: c& g' L/ Q& b$ B6 \F := QuadraticField(8);
% P; |' R+ ^" D: gF;
D:=Factorization(FF) ;D;
4 s. A. ^( v7 d. O. s7 OQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
/ Q- U% r: u& [! D7 w; b( t, }$.1^4 + 1
1 D( e4 k4 r( O9 B6 n- \5 bQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
0 M: u) h: u# b$ t9 m: A3 y    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
& D7 @  \5 m- G* K. ~) H% J/ c9 XF6 = factor(x^6 - 1)
" u8 G2 y" y5 d" R0 E1 j/ ]( pF6

1 m6 F( |8 X5 ](x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

( y& E" f4 _0 ?0 U& EQ<x> := QuadraticField(6);Q;
6 R8 l' F# c+ G" SC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

* Y5 l! k# b, VF := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
" `# F, _6 J0 p( A  o2 @  j' g1 ?Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
: j1 w; f6 k$ S8 b; nCyclotomic Field of order 6 and degree 2
& O3 p. \3 W# I$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
8 j( p& N7 O8 V' }1 a[
# r* `" }. N4 S( ~9 Y    <$.1^2 - $.1 + 1, 1>
4 U, d6 X2 M2 s$ B8 C% E8 g]

  l/ T- |- F5 }& F# \R.<x> = QQ[]
8 O3 Y( \6 b5 o0 Q$ M0 X3 hF5 = factor(x^10 - 1)
# O8 v  G) G9 R# H) D0 YF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

- A5 T6 w2 a& m, Z, v. qQ<x> := QuadraticField(10);Q;
- W% n" Y' S0 v' gC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
! A' F& H5 C/ g; w( _! L) g6 `
! a, L; ~; z& B& [3 N$ I* ?$ }F := QuadraticField(10);
8 v! P* u2 Q& D/ U6 y5 \F;
  q0 n0 d% L- D) RD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
" }) V# z4 L+ F) H& P$.1^4 - $.1^3 + $.1^2 - $.1 + 1
8 L0 v. `& Z* p2 s' fQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
" q, X  W5 w& C8 t2 U[
, K- s! B  B8 Z2 I0 L    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123