虚二次域例两（-5/50）

lilianjie

+ h$ v$ M, z/ o+ E3 Z" ~
' `7 ]7 N" `/ p+ G9 m% r) q* iQ5:=QuadraticField(-5) ;
4 Q, ?, e2 _0 F& |; [; T; dQ5;
7 a6 h  z% |9 Q2 r9 Z6 U: a% O
- K3 |, `+ E& @7 V# N9 f3 k( F% qQ<w> :=PolynomialRing(Q5);Q;
' K. j! z4 [7 {  i7 _7 E0 A7 FEquationOrder(Q5);
* k. V4 ~$ \% y; E2 EM:=MaximalOrder(Q5) ;
M;
& Y" o7 L5 K$ y: H/ C" T" PNumberField(M);
, B2 L! Q; v6 G8 U" o1 R& L* P! hS1:=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);
" m+ a" D( u) Z3 t3 p5 v8 T+ G2 FIsQuadratic(S4);
IsQuadratic(S25);
' k+ C% \- ~4 AIsQuadratic(S625888888);
Factorization(w^2+5);  
- C+ i6 j4 M( T5 d3 X7 l( u2 qDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
, v  c7 k( z$ {) i& Q/ U% A* GConductor(Q5) ;

" m9 A" Q: K( s# ]! r* pName(M, -5);
Conductor(M);
# [* A, T, r8 l$ R* i! NClassGroup(Q5) ;
) ^9 Z$ q& X! z0 M: I" qClassGroup(M);
9 d6 d- C) `3 W6 VClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

/ X" U* K+ U4 h; CQuadraticClassGroupTwoPart(Q5);
+ n& q( d$ v& p6 L- k6 }: yQuadraticClassGroupTwoPart(M);
( a2 f- F! j* K* VNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
; |& G3 n+ t$ R6 s. hUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
& m4 R- b5 ^% U' cQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
) x5 p' r0 ]% U" qOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
9 k7 v" t9 J7 [' [- H9 d; }true Maximal Equation Order of Q5
" |: Z" o$ _4 @0 t: xtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
5 {( E( \* a  O9 V% B- ^4 p5 btrue Order of conductor 1 in Q5
( ]+ n- {- j( M% f$ x; `' @[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
/ u- A$ {- P  u' s! m4 |1 f]
-20
: [( B1 c+ ?; j+ s
>> FundamentalUnit(Q5) ;
; N! r2 F' d) u: f                  ^
- d9 F; W0 `6 KRuntime error in 'FundamentalUnit': Field must have positive discriminant

6 D; e$ x  m, L# U" @
* ^; O1 y" Z$ \- M9 p( p>> FundamentalUnit(M);
                  ^
5 y2 @' E! [6 H7 y& S9 G" iRuntime error in 'FundamentalUnit': Field must have positive discriminant
1 U+ G( M$ q; J8 u* A9 o& f
20

' i1 n. b% ^5 I! T>> Name(M, -5);
       ^
) F. f  e/ u/ eRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

6 }6 X$ K! ^4 K8 ]+ F- T1
Abelian Group isomorphic to Z/2
& G; o1 A/ ]1 V8 d9 ^5 kDefined on 1 generator
Relations:
, a3 ~6 q* v& ?. T! V2 u    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
! f6 A/ [2 C# y; b6 }) U1 y. S6 _Defined on 1 generator
Relations:
: t6 q" Z) f  u+ r& i+ Q    2*$.1 = 0 to Set of ideals of M
+ F. W# \! R1 ~0 @& yAbelian Group isomorphic to Z/2
% U1 B! T% b' A5 k: sDefined on 1 generator
Relations:
1 a* ]" V3 ~- O1 H    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
3 Z- }& J) m: x9 t- c# G; ?Defined on 1 generator
* Y  ^. `! [+ t# a3 tRelations:
6 x: m7 W1 G5 g+ e    2*$.1 = 0 to Set of ideals of M
2
4 a/ L7 g! s3 w! _2
% K6 h& o' S$ r3 \# K  o4 Z. I; `Abelian Group isomorphic to Z/2
Defined on 1 generator
8 d4 a; E0 ], E! N4 gRelations:
$ u$ {0 t3 P6 R8 A5 s5 J2 {    2*$.1 = 0
2 ^7 @( \, ~6 X, t" AMapping from: Abelian Group isomorphic to Z/2
4 K' V" @' J: PDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
& @7 ^3 D1 N1 LDefined on 1 generator
# @, w0 [8 N9 f5 a8 {4 j% ~Relations:
- e& g- j! x* I3 X# c: n    2*$.1 = 0
4 a$ K& T0 I3 M, u; O% ~Mapping from: Abelian Group isomorphic to Z/2
# F6 z( n* W3 w7 ^7 M! BDefined on 1 generator
) @3 x) }" C; G- cRelations:
. K2 j* I; ?9 f) {* K* m7 ^5 J    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
" \# D/ w( L" t- ~Abelian Group isomorphic to Z/2
" G& P( I1 O* a% ^4 jDefined on 1 generator
5 Z; |9 C1 Z3 R2 H( w: }3 `. f) FRelations:
% O5 q; L& p- ?6 J- \    2*$.1 = 0
# @+ P; {' c; _; U' \2 S6 lMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
+ h6 i7 K' s4 v: o6 `' XRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
) c3 c+ R: Z" N. I: \2 f* r3 `false
4 j; i6 L! N) [' r) {1 Ofalse
, S1 d1 y+ j! {; R) V( `0 Q) e==============
$ F" C" R% C/ x2 E, G* c
) \9 W, [' l6 i; w& ?
7 G. f  p; U4 r- E% [Q5:=QuadraticField(-50) ;
Q5;
* W2 U3 ~+ V8 x: L5 m
: R4 L: N) l: c3 \. }% J8 \! PQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
9 J  ?/ H; O2 d: y# O; r9 ]M:=MaximalOrder(Q5) ;
M;
* q, r, G9 E. t- n, tNumberField(M);
. i: F" K2 G5 d" gS1:=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;
0 c, N2 R9 k( J2 t' H+ a& ]IsQuadratic(Q5);
+ c( y2 Y* m4 Z* PIsQuadratic(S1);
$ c9 L. ^* p3 R8 W  }- TIsQuadratic(S4);
IsQuadratic(S25);
6 r! ^+ p* C2 T; U- uIsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
+ J0 ~2 g9 M( |% s" l: bFundamentalUnit(M);
1 {+ N3 J: J! Z9 L1 JConductor(Q5) ;

Name(M, -50);
Conductor(M);
. C1 e! v2 ]; u5 u7 DClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
$ K, I6 @1 l+ p+ A0 c* qClassNumber(M) ;
PicardGroup(M) ;
/ K' t/ A; S( J6 ZPicardNumber(M) ;

- D( ]% [9 ^% ?4 O  UQuadraticClassGroupTwoPart(Q5);
1 v3 W' {3 _$ E* yQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
NormEquation(M, -50) ;
- {# e" q# V( \9 I# {! A
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' i" c5 S8 y" {% O& pUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
, K' J- `8 w/ `/ y( m  QMaximal Equation Order of Q5
& C) ]8 \; O. J% m) R' HQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
. p3 \; c9 B# z, ~Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
, w3 j$ J; O' R% l5 ]7 m0 s6 |true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
: c: \- X4 G; Y3 u% Z]
-8

! O6 j  ?9 `  ]4 B! B>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3 r8 q; D4 W" H  |  u+ E6 K
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

$ i- j& _: K$ V0 x" ^/ Y) V$ P  y8

0 Q0 s- |" @2 U% v7 z>> Name(M, -50);
4 j" y% q% i0 C4 z$ y       ^
" q; a& c1 ?0 x; u) FRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
9 x# w% ?' |" z, I6 N$ R
6 }& Z" ^0 j2 V1
* a8 n, q2 K3 r" GAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
" N. |% H# r6 p% mMapping from: Abelian Group of order 1 to Set of ideals of M
1 c$ V. q2 R% r+ C  P/ g6 k) o1
# G4 X4 c% S( K' S7 l1
Abelian Group of order 1
) N/ y& j" T/ L/ b( v; dMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
, w: x& p! i, z2 |6 oinverse]
1
( p# A6 x4 P0 }( l/ DAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
! }# h* Q0 O3 T5 m- L9 _Abelian Group of order 1
) L$ s& F1 Q. |Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 t8 `: I/ F- z+ I7 z) |-8 given by a rule [no inverse]
% j0 a, [7 u/ M; l2 E) afalse
false

lilianjie

看看-1.-3的两种：
  }: T& E, c' s( l5 ]9 T
5 {& ]# z2 y/ w! e) m. [/ ~5 ?4 J. ?Q5:=QuadraticField(-1) ;
; V3 B! n9 C0 a/ j' h- IQ5;

Q<w> :=PolynomialRing(Q5);Q;
+ f9 O5 l& T( l# i7 Q4 p/ b0 J0 \EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
# I3 P% `1 r0 r& w9 i) rM;
NumberField(M);
1 A& c3 Y. q* U) n/ 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);
; j# N4 H% ~" J3 G' F( HIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
* P$ B8 M& R5 }( Y2 d+ xFactorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
7 _9 b0 S1 n( _9 tFundamentalUnit(M);
/ }5 t. F% k5 S5 ^( h+ {& EConductor(Q5) ;
7 X+ T9 m! h% p7 K. P7 }
/ J( c+ \8 f  Y8 ^6 h: |5 `5 hName(M, -1);
4 A  C# ]" T6 w0 G, m9 VConductor(M);
, q, t) E# h1 S  NClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
8 v! i4 H/ G5 r# e3 K5 KClassNumber(M) ;
! {& y; x$ h  [  VPicardGroup(M) ;
PicardNumber(M) ;
  {! w) I& y3 X& [
* D4 D: ?* |$ G5 x+ oQuadraticClassGroupTwoPart(Q5);
7 L' X8 A$ u# ]- q( _$ @QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
/ Z) i" E. W* ~3 q  S  B& ~NormEquation(M, -1) ;
6 E' ?: S$ r' q  z1 H+ l
; `; m1 f2 l4 z9 V! JQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
1 I9 f! T% }: YUnivariate Polynomial Ring in w over Q5
+ _5 ?& F  {6 A) sEquation Order of conductor 1 in Q5
) o1 z, g3 a5 Z0 QMaximal Equation Order of Q5
$ l5 g$ L; m& W  o4 y, ~Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
' g7 p& F2 i# U3 N1 }# Y2 Ttrue 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
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
. `4 N! g0 g6 a. m! u[
: w0 k& `5 {3 N/ Q8 t4 h    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
, U1 L: \% ]9 c4 o" m: d0 L-4

" }" n* V0 Q) l. B  F2 J; X% V4 `>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


& W# [' X3 Y3 x" `, @: `  K% o2 I>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

4

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

/ D/ X- r& |/ G% P3 i+ b5 @+ Y1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
4 @4 E0 c% k* B2 M1 W3 DMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
: J0 q) R+ g8 j: A4 {- f6 gMapping 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
-4 given by a rule [no inverse]
5 k: M+ ]7 j! E2 r4 j4 TAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* I% U3 Q5 n8 z9 e# v$ p8 S+ B-4 given by a rule [no inverse]
& \1 x! ~. \$ f* G0 t; pfalse
false
: {( b7 W9 V3 B& h7 V: L3 e===============

Q5:=QuadraticField(-3) ;
Q5;

3 o, L3 M; C0 Q6 E# f# a. a. QQ<w> :=PolynomialRing(Q5);Q;
+ @3 x% |( L. q; j0 Y; f! ]EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
- N6 |1 y& @, P8 vM;
* R* m5 e( L; }3 l* Z# qNumberField(M);
8 s: \5 z/ y3 l3 W6 [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);
" Y* l" E8 z- }0 }7 o. yIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+3);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

9 {; m+ c5 q8 X* t4 t! |7 m+ G( @Name(M, -3);
Conductor(M);
) a# `4 P8 W2 I- R' DClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
4 r- x7 ^3 _3 S! n& \PicardNumber(M) ;
  \% [* }6 i; g6 u
- n- d) @. U8 m* Q5 `7 sQuadraticClassGroupTwoPart(Q5);
, z& R& S. |+ r: pQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;
: |+ P* M6 ]! N( O8 }
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
: H7 V4 j+ G9 H" r/ l4 GUnivariate Polynomial Ring in w over Q5
2 _5 n- N" ^6 xEquation Order of conductor 2 in Q5
9 I: k' B5 F; K0 {, w! U1 sMaximal Order of Q5
1 j4 H3 N& v9 i4 m- nQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order 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 n! L6 v& V" Q/ w& e4 L) H, J6 Vtrue Order of conductor 625 in Q5
& U5 z4 H0 j% o  Atrue Order of conductor 391736900121876544 in Q5
[
, J0 e( \# v. \    <w - Q5.1, 1>,
. ~' O6 j8 L# N4 a, v; Q% A; {& f    <w + Q5.1, 1>
]
-3

/ b% v( H! d+ m  E( x. v9 m>> FundamentalUnit(Q5) ;
0 A7 J; Z- S- G                  ^
. d* I, |: ?+ ^. \, LRuntime error in 'FundamentalUnit': Field must have positive discriminant
2 R9 b, Q) ]' F( ^" R2 U/ Z" e# U
; o4 t  |& B2 y7 \9 n) s# `
; h' _1 A, c! L6 Z; \8 U  {& k3 X: ?>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
% T$ n+ b/ k; M/ K
" I" I1 v+ f* u. H3 E( _3
  R0 }* _- ^+ h* h% X1 T
6 T7 d0 {5 V, R6 V>> Name(M, -3);
, D7 ^$ e1 W+ c  N' s1 t       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

% W5 }' l  ]  P  W/ ]& ]0 h* @9 j! n1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
3 `7 t& |4 z; a$ L1 w1 |) KAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
8 |+ o* F$ l+ q+ Q6 N1
/ o; ~( {2 ~* U+ ?0 a5 [+ w$ c1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
, \3 ]' M0 g) V. y& x1
Abelian Group of order 1
! y+ V+ @) C+ ]Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! E2 ?' b  o2 I, c( f0 k-3 given by a rule [no inverse]
/ f% t+ T3 }7 L7 P- X& O) PAbelian Group of order 1
; x' s+ q$ R. E- iMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
0 D' Z+ K& g) r-3 given by a rule [no inverse]
  I$ f' t+ ^7 O3 f+ r, Ffalse
3 B" ^$ s$ a, t6 p; l$ {* S' Ffalse

孤寂冷逍遥

lilianjie

- B+ i) s% t% v. r/ `* }$ ]( n
Dirichlet character
( y1 G  X: f% j2 p2 VDirichlet class number formula
- U: j% j, Y$ b8 a' }0 g
7 [; q$ ?) L; f. q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
8 h  u2 H" P! B$ _2 {
) b* p  b% N; G% |$ q# _-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
  `) O# u) F# m& I7 q& gh=-6/（2*3）*Σ[1*1+（2*（-1）]=1
$ \/ w7 ~5 l" a5 x! n4 Q' F
-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，

  ^: P* m5 k- E% g5 F" M

' O: g# }  ^; }h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
' c( o0 E6 s& f6 |2 C  z/ k
: a1 q7 \' x! _- [0 Q

4 J/ c+ \- y! i) }3 w6 D-50时  个单位根                          N=200
  x, f) H6 U2 d6 c+ X: f

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
ClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
0 F7 F5 d) E0 m) G( F& n. h5 tNormEquation(F, 7);
; m/ W& G; b% e0 ^A:=K!7;A;
B:=K!14;B;
% \1 z7 s) E( P1 W. j! A0 V  aDiscriminant(KK)
7 X$ z- H  k' P, `- S; E8 R
7 ^: t# e' j; W1 [Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
1 M; ~2 B/ {* V8 ?7 V" T28
  n/ t: }* a5 p+ OAbelian Group of order 1
8 p" z; H& _2 j  s8 d( X5 T& y9 ^Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
$ ?! i* B+ F6 U" tRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
/ O. b; D7 B+ a: V    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
( J% N: [# }' o+ p) l* w- tfalse
7
  |! f" T9 W- `. r14
: \1 l0 a; h8 U- I. o$ b28

lilianjie

! }* i' Z6 [# N* W

# g: k. _( c& d0 W6 e, h
( F! y+ u* e# T2 G, i: c
1 c2 m: B/ p8 @; e# T
分圆域：
% B1 I$ R3 m) Y5 C0 [% ?1 Y3 e( i4 hC:=CyclotomicField(5);C;
4 |! t- W( c: U6 d" f# pCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
! v' |. }7 [% S. S' V' ?7 M3 kMinimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;
1 B' O7 f, {- \% |
CyclotomicAutomorphismGroup(CC) ;

Cyclotomic Field of order 5 and degree 4
) F5 l; N6 M" m0 x$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
! Y3 }( Y" i: ?) d$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
& M5 A3 u# J) b% H. G8 RRational Field
0 V2 w2 y: c& Z* fRational Field
Rational Field
Rational Field
zeta_11
# {4 e4 L% N1 f5 r7 Fzeta_111
123
7
) H' m6 P) c  i8 j7
* U" p5 P' {1 S: m0 Q7 i- ?Permutation group acting on a set of cardinality 6
# @6 F' c1 A9 l: M2 W9 ^Order = 6 = 2 * 3
9 R* m7 J4 O+ r8 r3 L    (1, 2)(3, 5)(4, 6)
8 L1 F+ ^4 o- p6 _  N7 @    (1, 3, 6, 2, 5, 4)
3 G) Q& G" y* q; C  a; C; gMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
  {! d7 B6 e* H/ w2 N7 f' y, a+ ~Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
) J; S8 }7 B# O$ lCC

lilianjie

lilianjie 发表于 2012-1-9 20:44
: ^1 D( }! }% Z4 x5 D% Z* x# M分圆域：
! _5 d/ Q4 P, [/ E* JC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123
0 b& D7 \5 a7 E8 `5 b- i3 E
0 j! X, k5 I7 CR.<x> = Q[]
F8 = factor(x^8 - 1)
9 k- e- v' J- R. u; L- {. _F8

  J3 k2 ~4 d8 e) W/ _, s(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

( w+ R4 P+ t& J+ M$ S% r9 j7 @Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
: o4 S) O( [: L* H+ \2 ]9 I! G: dFF:=CyclotomicPolynomial(8);FF;
. u/ p; n9 [/ E
F := QuadraticField(8);
* }2 l3 C* F% C& o+ H* }2 UF;
. z" L+ b3 l# A. p5 r* Q& t/ w; LD:=Factorization(FF) ;D;
6 o9 y3 |2 z& x0 zQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
0 z& s( J) l! E& c; D1 G6 |Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
5 Y" g9 D9 i" s( k( u4 N/ Z# @    <$.1^4 + 1, 1>
]
6 p7 K. |3 V! _, M& G. q
R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

% W! m/ {9 E+ o) wQ<x> := QuadraticField(6);Q;
& h; D2 W. o* X( S2 n0 B+ EC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
0 ]" b. H: K$ A3 Z7 o! X/ m
' |# O' I5 e  {! R2 B) T- }F := QuadraticField(6);
# x0 G* ?' g( G7 _F;
& J9 X5 [) e0 a( b: Y% I! GD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
8 K8 q% \+ k* B6 gCyclotomic Field of order 6 and degree 2
( s* M: M* s- ?, X% V4 W8 l: |7 p4 B$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
5 y. D5 |9 G8 H0 I2 q[
    <$.1^2 - $.1 + 1, 1>
! }4 ^# T' P7 P+ k. K5 V]

! s+ K& o4 A0 S6 z: LR.<x> = QQ[]
F5 = factor(x^10 - 1)
3 W5 R% t1 i# y8 r8 S) B' {F5
(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)

$ M* d& A  y. v  c5 AQ<x> := QuadraticField(10);Q;
; F+ k! H' b8 }3 d# FC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

  e# w8 N( A: q9 B( yF := QuadraticField(10);
F;
% W+ R  E/ k1 Q0 cD:=Factorization(FF) ;D;
7 h9 [3 G. f) F5 {Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
( J$ W$ S( n  E' ?+ C' SCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123