虚二次域例两（-5/50）

lilianjie

' X2 t. O( K5 A3 bQ5:=QuadraticField(-5) ;
  P; S1 K4 d! Q. ^: L$ s7 pQ5;
  x1 z5 ^% }0 P8 p: t5 B0 B8 i
- o  C+ e4 M8 G; L8 `  JQ<w> :=PolynomialRing(Q5);Q;
* E5 l8 \- C$ e! ^* dEquationOrder(Q5);
6 ], P& `4 ?1 y7 e" T4 mM:=MaximalOrder(Q5) ;
# x1 Q( \! J, ?# }% QM;
' i, T- [) m. G  tNumberField(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);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
4 k. N; o% ]$ _$ eFactorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
; f( m7 J$ ]- d( y4 |6 JFundamentalUnit(M);
7 x  o4 ]# @  }3 {, t% cConductor(Q5) ;
$ |7 @( R7 J! l! Y
& {1 l6 i0 v: g, B7 o- sName(M, -5);
Conductor(M);
0 H% C2 i5 b- l  pClassGroup(Q5) ;
7 R2 a- c' u1 [0 ^7 x9 U1 p# ZClassGroup(M);
' ?: z. O8 t& t$ k$ C# C( J& t5 A0 JClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
4 Q. v! v1 Y) `# {PicardNumber(M) ;
8 t$ N% c- H0 A$ J
/ b6 Z( o6 _( _5 c! |+ J( rQuadraticClassGroupTwoPart(Q5);
, R0 T- ]. d/ }& T: QQuadraticClassGroupTwoPart(M);
( T2 r0 g( b! [  |: `3 l; d$ z; [* HNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
, J3 J; c7 K5 l( }6 Z+ N- oUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
: ?6 T# z; {7 m: a" t% h+ pMaximal Equation Order of Q5
7 Z$ G; R: G. M7 ^Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
' Y+ C6 \8 V, I1 \true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
! v+ `( i! d  k" A$ V3 e( Atrue Order of conductor 1 in Q5
, r' L4 J  b( F2 ?! ~7 }[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
) ~' R5 k' N' ~  c! j$ y# p]
, a' _6 ?4 [% Y-20
0 T- n7 w- C1 ~- M: C
>> FundamentalUnit(Q5) ;
+ N9 ^, u: e. ~& @$ C8 }                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


  u/ D: N; s2 J4 f>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20

7 U1 r7 K0 J7 M+ ^1 c+ f0 u>> Name(M, -5);
2 v4 L5 |; J5 \( {3 O0 [       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
Abelian Group isomorphic to Z/2
Defined on 1 generator
( f  ~) o+ p8 F& s# MRelations:
    2*$.1 = 0
2 F5 a9 W. @4 \* E; @* OMapping from: Abelian Group isomorphic to Z/2
6 \4 _: w/ ?* ^4 N, DDefined on 1 generator
Relations:
+ l& F3 g' k' m  k. z    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
6 B) U# b8 X. q: v  ~+ mRelations:
9 b+ K, j8 r6 @, Z, Q    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
( s: C3 l6 Z. LDefined on 1 generator
Relations:
4 [0 K8 g0 E; D: K* }3 o& \    2*$.1 = 0 to Set of ideals of M
2 w, f+ v& p( a. k5 m3 A2
2
. \% h0 D  {+ _/ E1 K4 B; f& kAbelian Group isomorphic to Z/2
, ^7 M/ p% J7 a7 X; A3 ?, R  M( O6 |% R6 JDefined on 1 generator
+ _$ T3 q1 p( g* I# O6 hRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
3 t9 p8 Z# O% k  FDefined on 1 generator
) t( t  }" ~9 W0 K2 |. DRelations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
( m4 J+ A5 P- W( K" m, N4 v2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
5 f$ y* x8 E+ N' H/ W8 E/ JMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
6 v. e9 q% N, C$ A- F. }  j    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
5 @8 g# i" Q# M. ~inverse]
Abelian Group isomorphic to Z/2
; x, ]) P& X# d. Z9 W: L8 |% [Defined on 1 generator
Relations:
6 y( ^5 Q% M" P8 X3 q    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
0 ~8 ?$ z' O  }. `4 B0 p# sinverse]
% g: `7 B* r% k1 W) b& }& u; P  lfalse
false
==============


Q5:=QuadraticField(-50) ;
Q5;

3 [0 A( Q9 C1 _$ Z5 `  H) w( x" D; eQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
4 N" ^& }3 X% ]M:=MaximalOrder(Q5) ;
5 ~& E6 l0 N! D4 D; h7 }M;
5 N- b$ p6 h$ L) @8 cNumberField(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;
7 I3 j" u9 Z* @IsQuadratic(Q5);
IsQuadratic(S1);
! `( [. c- j/ r# t7 [% ~$ vIsQuadratic(S4);
6 e) t* I  g" w9 q' ^4 iIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
+ D! u6 ~9 b. [8 D# ^Discriminant(Q5) ;
FundamentalUnit(Q5) ;
$ e9 U1 w3 t+ Y3 \& C+ G, p# HFundamentalUnit(M);
) S2 @! l3 W: y. y1 z* yConductor(Q5) ;
) e' C( E2 W$ l
- s# J. G+ X" \# d2 h) P5 [! h5 J) l4 uName(M, -50);
Conductor(M);
. S) z% ]) w- @: ^# z' @ClassGroup(Q5) ;
ClassGroup(M);
7 z2 h: ~+ ^. \6 |9 vClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
# z" ~  N1 P" @) c8 w3 O: \* Z$ i
QuadraticClassGroupTwoPart(Q5);
: g  Y5 Z6 J3 DQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
4 |- F% M, \. Z6 mNormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
* [. {5 ~; a! V: o3 s: C, h+ w) QEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
% l$ ^7 Y" F6 J6 YQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
5 d: T! }& F; p" L1 dOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
1 i9 H+ [" r9 s. j: ?true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
2 a4 f% F  X$ [6 N5 w# c5 Y$ l[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8
' R% y/ A3 M) p& z
>> FundamentalUnit(Q5) ;
7 P9 D+ Y1 ^7 Q+ B+ i) e, s+ u! E                  ^
/ l! h; i% @' B) s5 ^Runtime error in 'FundamentalUnit': Field must have positive discriminant

1 R& p5 @3 v+ R
' k  c: |3 H: D  w' _>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8
# v$ ?# D1 u. f1 F4 T" x
0 |' j7 {5 j0 c- O" j* q: F* U( f>> Name(M, -50);
- d9 V9 D! f# s7 {; D       ^
5 F- y) ^1 h' r( Q; |0 D5 ]" {Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

1
$ A" a3 X! \5 N% U; ^& I7 y: pAbelian Group of order 1
* O6 v2 ~( p. ]5 }Mapping from: Abelian Group of order 1 to Set of ideals of M
: N% Y0 n" @6 v* u7 WAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
7 v+ i: _" N* k! b1 C1
1 D) a3 [1 t7 `+ F8 e6 fAbelian Group of order 1
: N6 O- z3 F0 E% B0 dMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
5 o# ~3 }; E. r' Ninverse]
1
0 h% P* R) u3 RAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 ]8 s& J9 R& q2 x6 C: r3 j+ ?-8 given by a rule [no inverse]
+ f* v: f5 B; o6 R6 e+ d  Y  UAbelian 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
false
) g9 p8 T2 c5 Z, z. L/ T

lilianjie

看看-1.-3的两种：
& t8 B/ q; B. H+ @4 |
- {1 c7 g- ?6 M6 i% q1 I) V" DQ5:=QuadraticField(-1) ;
8 n% D1 a. {' ~& G/ V% a; G. B8 bQ5;
/ X8 A- ?( A. t/ \3 |6 E0 d% u! u
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
! l& W" b, R7 V1 C' EM;
% {- p, `& A, KNumberField(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;
! [7 n4 `  e4 R1 Q9 [IsQuadratic(Q5);
% E4 h. q) J/ ~% @, t* q) Y% r: n6 jIsQuadratic(S1);
- ]- K6 C! Y* ?$ \2 YIsQuadratic(S4);
+ R, B9 i. H( ]7 S7 B: lIsQuadratic(S25);
/ W2 j" t- n) ]" h$ n1 d. R0 dIsQuadratic(S625888888);
Factorization(w^2+1);  
$ l* P; I! I+ O! r! p+ N2 }Discriminant(Q5) ;
: A+ A7 [! ]2 u  r! wFundamentalUnit(Q5) ;
: A) n+ h; q; y% V. d% T! O, eFundamentalUnit(M);
Conductor(Q5) ;

! y% a6 |: q( [8 J" \7 ~Name(M, -1);
3 q, n7 H' b; m# e3 yConductor(M);
ClassGroup(Q5) ;
1 E1 y  g' i3 P! Q* x1 {1 ?$ |ClassGroup(M);
: L) ~% o& q- @; M3 F0 i! kClassNumber(Q5) ;
ClassNumber(M) ;
  ?3 d1 N: `' q8 U7 rPicardGroup(M) ;
/ A8 Q& i- Y+ f, g9 ^PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
. R# l* g. h) ]9 ?0 l2 |1 fNormEquation(Q5, -1) ;
6 j1 S. V& q6 W; cNormEquation(M, -1) ;
. o5 W$ D* {- U, A$ l
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
" r' r: q# ?/ fEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
, p/ l/ L" Y, v" X. l8 z( ?Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
* T' |$ T0 F% S: i" KOrder of conductor 625888888 in Q5
. S- N. k1 N0 rtrue 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
1 C& U- c! p8 t  ^  k1 D4 M: k1 j/ Ctrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
3 c; [( L1 r* Q[
9 T7 n. Z( j- g" S1 N    <w - Q5.1, 1>,
' d0 p* Y% q. t1 |8 `    <w + Q5.1, 1>
]
-4

6 V: ~+ \; r7 P' j9 j3 A>> FundamentalUnit(Q5) ;
0 x: U* v. K: q* h                  ^
/ B1 {. R4 ^% D9 yRuntime error in 'FundamentalUnit': Field must have positive discriminant

0 n3 X' U* X6 u+ P
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

) X  a) x- `# p8 F4

, T& N2 h; S" x7 e# a' V) l>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
9 _) M, S3 @* D
4 a1 C- Q  h) g. R# B1
) ~/ y: c9 {  z1 @" x) PAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
5 |6 s5 d, K* ^2 ?/ ZAbelian Group of order 1
# w% }+ ]* g" R6 R7 p7 iMapping from: Abelian Group of order 1 to Set of ideals of M
1
+ |- F( o/ ^5 k0 p. K7 c1
1 T2 c  L5 l0 E' `3 F) HAbelian Group of order 1
- t( t* M, n  t% k4 PMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
4 @/ V5 j0 [7 Kinverse]
1
Abelian Group of order 1
$ H3 Z( u3 n2 Z6 MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-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
2 u4 `6 |, u& D  l* J, K& C. ~-4 given by a rule [no inverse]
) A" Y4 v7 ~0 X. X6 ofalse
false
% }) X* `& c  i6 Y% A===============
2 k6 d% }; W& |* {
Q5:=QuadraticField(-3) ;
; G/ J$ J( V' i4 E/ ZQ5;

Q<w> :=PolynomialRing(Q5);Q;
4 r6 |4 e! N  tEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
" D! r7 h! ~( u0 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;
IsQuadratic(Q5);
' C8 H  ^0 s! nIsQuadratic(S1);
( m3 T' t6 h  r" u, x& F/ p$ v- G& QIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
+ e, s0 B- D( V6 H: tFactorization(w^2+3);  
* ^, A/ T: B, ?; t: W8 j& K) N( GDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
1 M, b' ~  X% aConductor(Q5) ;
. X. M. R' n' M) ^. c' H4 k
2 h+ P" `8 d9 s" qName(M, -3);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
# N! n3 H0 w, t- V% ]2 fClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
2 }- |; i  m" S0 Y) Q3 h
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
+ Q' W! }; j2 s3 S- J% i) K5 KNormEquation(M, -3) ;
" u4 X( q+ s0 {& ~0 A
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
: [4 X: W# J  TUnivariate Polynomial Ring in w over Q5
: r9 o1 q: {7 {5 cEquation Order of conductor 2 in Q5
Maximal Order of Q5
4 }  q2 o# u  B* \. xQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
: E5 T' b+ J" lOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
+ n6 |  E6 H- _% B2 s6 u# U: R& _true Maximal Order of Q5
6 A8 R9 z# E. S; r# Ktrue Order of conductor 16 in Q5
- s) K" M! K  Y7 V8 y3 @5 @true Order of conductor 625 in Q5
- f9 B( I  s8 v  Strue Order of conductor 391736900121876544 in Q5
: n( y! v3 i6 T/ s[
) R- Q: b( Y' N    <w - Q5.1, 1>,
    <w + Q5.1, 1>
* J& Y$ X9 m+ b- o, q]
-3
2 G. G3 b9 q8 W7 N8 i, S3 I, O" H
, V  ]0 a* A, ~+ T) _0 {' H>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
) Q; R' L& B3 z/ ?1 Q! f: _# h( |, |

>> FundamentalUnit(M);
9 L/ K3 M* r6 C. s: s                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
7 m8 @# G! t# `6 g8 t' M$ e1 H
/ r. Q* S) P4 N3

>> Name(M, -3);
# I  D* O, |, ^) M/ G. H       ^
3 R3 f3 |4 r% ~8 v! GRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

1
) O1 b, @7 U/ C- ^Abelian Group of order 1
0 B; Y- @- N  s) n/ f2 ~Mapping from: Abelian Group of order 1 to Set of ideals of M
4 v' u4 Q. V5 z( AAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
+ k! \; u1 G. X* N7 a1 l, PAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
. v5 f4 |- N1 H7 e* R- \1 L" ]inverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
Abelian Group of order 1
# ?3 |/ @; B, J* W5 \+ T6 R7 _Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
3 f7 }  }& y% ]-3 given by a rule [no inverse]
8 K! E2 F4 v; ?% s7 ~false
: _5 x6 l4 r  F7 g7 y3 F: Ufalse

孤寂冷逍遥

lilianjie

Dirichlet character
7 m. c- i/ E) D  @) `( [/ Q. |Dirichlet class number formula
, c8 M0 @- N* N% A- C8 e/ J3 X
% s9 B/ U% t$ z8 e, }; j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
% z6 x9 Z; c: ?9 ]; z$ P6 s0 C
-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，
$ }/ `6 Y, q0 g3 W: E8 r/ e$ g. |


) H: A5 G" I7 }6 R+ x* Sh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2



9 J' j1 {+ X( x9 k5 p-50时  个单位根                          N=200
8 L4 E/ |$ j5 N$ ^2 ~

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

2 [% l5 s/ f  q* D& jF := QuadraticField(NextPrime(5));

5 o2 N5 M" R& h5 PKK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
3 Y+ U, N5 f7 b* T8 U& {" {% v" G& ZClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
+ D* @( h" M: o& ]) o  {A:=K!7;A;
/ m7 H# N& g! k5 `1 y! ZB:=K!14;B;
Discriminant(KK)

; a! F9 Q. [2 I/ }6 N0 d/ fQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
7 p) `  T; h$ q4 M1 e28
Abelian Group of order 1
+ s9 x5 i  J7 L: P# v; `Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
, F! X7 t1 T( ]4 y; j$ }    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
9 h4 g" G5 J& a* k5 M% ~. z    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
* a2 D0 e4 L  D! Sinverse]
* m" L7 q) h$ N8 b  k8 m# Yfalse
, p4 L) b, @" E7
14
, h, |7 n5 X5 j' U4 V9 q; b5 N# S  n28

lilianjie

( [5 P  @  T# w# O2 j



( f' F: G! B9 f7 \4 x
' Q& R8 z6 X! \& X! e5 l: z8 Y- v: `
" _9 g0 u$ o9 ]0 y) F
7 R% M+ P2 H6 J5 c# E4 _分圆域：
! h' }, y! ]4 Q7 _7 h) vC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
! l" D0 S0 A7 W1 {3 r' \& P- ICyclotomicPolynomial(6);
; d1 c" A. ~/ p( _* xCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
* F/ I' ]  Q, u+ i* e9 }MinimalField(CC!8) ;
# ?5 `8 ~* |% C$ Z1 H3 iMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
5 G7 t) R( z( g0 r$ U2 ~9 B, [Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;
$ h% Q7 W) ]+ T' M& H& ^6 R- p
CyclotomicAutomorphismGroup(CC) ;

  Q6 l4 K5 f' ~Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
7 S6 P/ @5 P+ L: UCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
0 y$ f& j9 n4 \- S3 UCyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
2 g5 o' X8 H9 `. \5 J5 dRational Field
. g' o; C, l5 @3 ]* Q6 E/ P# F, D; {Rational Field
Rational Field
5 @/ x4 l: q! V9 y1 H5 g8 A9 h2 WRational Field
zeta_11
zeta_111
+ w( h( y7 K- W- K2 m123
/ t9 n) d9 T+ r, e7
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)
* C& z5 V+ Q  b; yMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
4 I4 e( U0 ~* i0 `- h; FCC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
6 [* o1 t( M2 J3 G8 h( ~Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
# p* ]+ _- L; @4 X8 ~; NCC

lilianjie

/ n# r2 ?  u; g" p9 s, _

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
" D, V) b/ n# y' {: F, VCyclotomicPolynomial(5);

" G) ^$ W5 v; n% r' ]) ~
分圆域：
分圆域：123
) ^1 f3 r1 L- M: H/ G0 n7 `
R.<x> = Q[]
F8 = factor(x^8 - 1)
$ _. `+ Y, b' u4 h2 M5 MF8
2 f" U4 a9 f2 _: ~3 }3 s/ Q2 ~5 {4 `
- r/ g& P5 q! T8 w& r7 E(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

+ L3 F+ z0 I$ zQ<x> := QuadraticField(8);Q;
- `% e8 P, f3 N* W3 c# A4 CC:=CyclotomicField(8);C;
: \0 O2 D; D, J) _FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
5 k. k, y- T! M$ j+ d8 g+ B6 eD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
# m1 }+ _3 B/ B+ g+ k6 r$ ?$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

1 X. A# t* N) _/ L3 g! ^R.<x> = QQ[]
' \, L7 H) O9 Y! h; t- [: ~F6 = factor(x^6 - 1)
) r* j) j4 a% H" g! T+ E. T: f- \; G# KF6
6 _; f1 \4 O; L% R/ O" B2 X$ G
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
5 G' q. g- Y8 j( |C:=CyclotomicField(6);C;
, O7 A: w6 y) j% k  I1 TFF:=CyclotomicPolynomial(6);FF;

% J' Y0 T/ S6 |; R$ jF := QuadraticField(6);
F;
8 I) @+ k5 h& z6 C* aD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
' Y6 z: `0 r5 o! Q- r- DCyclotomic Field of order 6 and degree 2
, ?; `6 `9 f) Q6 n/ {! @0 L$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
7 U* B. u+ L* L]
# J  m! Q) }' F) ]" x0 s" @
R.<x> = QQ[]
F5 = factor(x^10 - 1)
5 w- i- ~' u: e# C; @5 oF5
(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)
- A  C1 A' T" e5 ]2 J2 t; _
Q<x> := QuadraticField(10);Q;
! d; l4 t0 y, v' \C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
6 M! J2 P* [& z6 Q6 \) P$ r4 ]5 V) H9 FD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
" O1 E* I0 f( g/ F  R5 jCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
1 e6 B/ X7 S- f! {: n! C1 A) ]Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
/ P5 u/ t* _' H; U9 a3 [    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123