虚二次域例两（-5/50）

lilianjie

$ K+ Y/ q7 r* [: W* v! Z: j8 Q
- x; \1 z; J+ r& P7 {1 B0 W' kQ5:=QuadraticField(-5) ;
3 W6 I/ ^/ s. }* b6 O2 V0 h1 X% UQ5;
9 r5 n0 m/ I% {) D4 n: \! _* O3 T( G
Q<w> :=PolynomialRing(Q5);Q;
" D& @% Q: O, d8 K8 ~; rEquationOrder(Q5);
6 O/ S$ p. @. i6 I& E& R. ^9 G( FM:=MaximalOrder(Q5) ;
M;
NumberField(M);
2 A) \% {8 z; }* _! IS1:=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' Y/ T' w* A- L! V: `/ l1 UIsQuadratic(S1);
: F: i' H  x1 a) P3 ^IsQuadratic(S4);
IsQuadratic(S25);
$ z7 ]' x6 M4 v. _5 ]$ EIsQuadratic(S625888888);
Factorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
5 k9 p) l8 U8 `5 m' B- {8 \  sConductor(Q5) ;

Name(M, -5);
Conductor(M);
- U0 q  }6 ?  c# F! [2 L" xClassGroup(Q5) ;
ClassGroup(M);
& _$ }$ ]6 K+ P- xClassNumber(Q5) ;
. @& l- }% ~' \ClassNumber(M) ;
1 |: o' e. L; {7 m2 ^PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
6 W9 G" e2 }% A: f6 }* nNormEquation(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
Maximal Equation Order of Q5
; a" E5 ]4 d, M7 {/ u2 X1 VQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
( V" i" [1 V+ a' Ftrue Maximal Equation Order of Q5
* r: n. F% w& p% T6 Mtrue Order of conductor 1 in Q5
, I- o+ m$ @* ntrue Order of conductor 1 in Q5
* |) Y" i% }$ ?/ h" W' _true Order of conductor 1 in Q5
7 L) u' o* q0 ?, P& @& Z, V[
1 q- ?( i3 {" V% Y/ F4 s    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
1 M8 Q) N$ `& j-20
. b% }; v! A3 d- L8 \
* z9 W+ F# ~; c  N>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
: ^) ]; @/ ?1 i3 R

3 |' g, I. `- X7 J& \>> FundamentalUnit(M);
) T1 A+ V1 n% C3 z/ D                  ^
9 ?5 Q. X& _# TRuntime error in 'FundamentalUnit': Field must have positive discriminant

. I; G( E, N1 x; r20
5 N! h2 h6 x  @) H# f- c7 {/ }0 g& H- T
>> Name(M, -5);
       ^
3 N  s+ z: a3 b; v9 o# K; p. YRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
$ O8 V/ h+ u7 a, H* L3 \6 J
1
Abelian Group isomorphic to Z/2
5 g, O  j" g2 D! CDefined on 1 generator
Relations:
    2*$.1 = 0
6 z7 W% s3 g' D, C& ]+ r/ oMapping from: Abelian Group isomorphic to Z/2
! |7 Y3 Q! {) u! G& ODefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
) n! K3 n9 ]% w) i5 I. vRelations:
9 t9 e' u2 ~% a) q4 F0 Y. \2 C) Z- z" ]    2*$.1 = 0
" H$ Y6 |' u8 E9 z$ \Mapping from: Abelian Group isomorphic to Z/2
8 Q" [* N1 O. i( z) S2 S7 jDefined on 1 generator
; a+ [" U, R" t. t$ ARelations:
    2*$.1 = 0 to Set of ideals of M
; ]$ b: x( T2 q$ n$ E1 I2
; H7 F2 N& l# u% m/ m& U$ O/ h2
Abelian Group isomorphic to Z/2
Defined on 1 generator
& n3 ]5 E, `, w& O( MRelations:
; O$ |* _+ ?$ A; j    2*$.1 = 0
9 U" `& x: k; Q' P* T( ?6 v6 gMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
; c9 @! M: a- ?$ {2
9 f" k9 Q, v% o' h. j8 h( YAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
# Y2 p' W2 s: L! AAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
( h5 u) p# ]6 F; p9 t+ n    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
+ r5 ]+ P9 Y6 X/ A+ LDefined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
1 }+ w0 v% N# d( o9 k3 o' @/ Iinverse]
false
4 E/ j; l2 \# l8 i2 hfalse
==============
$ k# r- t# W' {* {1 @9 C/ H

Q5:=QuadraticField(-50) ;
  X0 \  v- U& t0 T; z6 LQ5;
7 A+ k, m$ D! Q% w4 P2 x
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
2 h9 `, ^0 x! i" O6 _# ZNumberField(M);
% b* a8 g/ f+ B; E6 cS1:=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 y' Q1 x1 Z" r& a2 OIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
; E; K- u+ a4 n; @IsQuadratic(S625888888);
# t' A3 Q; s- |  D" Z/ mFactorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -50);
7 \* E, H% `8 R7 R- y- aConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
/ b; V/ a$ \8 F3 g5 cClassNumber(M) ;
6 {- n! }0 S( ~, s' fPicardGroup(M) ;
PicardNumber(M) ;

" F, x& ?: j/ L2 z+ T3 X, q* vQuadraticClassGroupTwoPart(Q5);
& e1 w! K, d: ~; lQuadraticClassGroupTwoPart(M);
! o( x( T2 I2 U: ]8 ?$ vNormEquation(Q5, -50) ;
NormEquation(M, -50) ;
/ ?  j$ N# z' n; ]0 T" Q! X9 K
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
5 x' \0 P7 p$ a/ P: EQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
. C* \1 C. j( t: M- P2 }true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
+ S4 I& U! h9 l( N9 {9 ?5 {5 a% ftrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
' H) N2 u0 o: j% f: [- N3 p2 [; I) Ttrue Order of conductor 1 in Q5
; I5 a' Y# s% l5 s& @3 K' Q9 K+ wtrue Order of conductor 1 in Q5
% w2 q- U9 b" ~% }: Q+ z4 Y% Z[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
. V8 v  P1 }! U" v5 M$ v6 s]
8 w4 b8 ?+ }0 p* v  i% C8 f-8
7 o6 [% ?4 s8 w; T- M! w
- X1 Z+ ?0 H' T>> FundamentalUnit(Q5) ;
# C% `( ]* A, R$ P* A                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
; N, E  y2 O" ~" o: m$ y- \% ~
) R7 J4 A3 _7 W4 X& q. j# B7 u
# z3 @2 H. ?4 h2 I* u- g3 {>> FundamentalUnit(M);
& F6 m) T, o/ G                  ^
. C. h; s# v3 n2 `Runtime error in 'FundamentalUnit': Field must have positive discriminant

! E( h$ j) e, Z2 {; i8

>> Name(M, -50);
* C! N3 t! P- f. F7 B! d2 n0 s! r       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
% u: F1 `2 t- y, Y0 e* L! P
' J# Z/ O% N3 c- y! A+ l1
6 r" `* Q5 h8 E5 OAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
% }$ t+ E, v6 xAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
8 \" G6 b, c5 ~' g* w8 C1
- ^' l# y$ r/ H' s8 ]) G8 eAbelian Group of order 1
& R6 J- H" m" M6 C/ C% EMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
. r+ ~- n. `* y$ b, R, f" O# Yinverse]
& `: A, h* }5 e1
Abelian Group of order 1
/ ^8 o9 q4 N, }( k6 G2 I& Q' hMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
$ i6 s- m! R( w3 I2 L7 bAbelian 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
$ e" a1 M2 P1 h7 e; J# i3 Gfalse

lilianjie

看看-1.-3的两种：
0 f' }* X( j- P/ e  @
. l7 {$ k& s0 e$ M- A' q/ Y. SQ5:=QuadraticField(-1) ;
Q5;
) D- W$ ~* z5 b* T0 L
  H2 G+ y" q% ]5 L1 f9 sQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
$ K+ q' n5 m4 X' xM;
/ X6 n. r9 A. @$ K* Q) Y4 `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;
1 t, d# d0 [6 BIsQuadratic(Q5);
! }$ d5 ^# l9 Q5 q  DIsQuadratic(S1);
4 i& R0 A9 j) \/ A7 [IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
/ {& Z0 R1 F' b) P+ LFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
# Q$ k3 d/ H, s, [4 ~7 l) d
, ]  h+ V) }7 v6 J0 J+ U8 bName(M, -1);
1 E! x8 U6 F1 J; gConductor(M);
6 @% y1 k7 \! {- I% `ClassGroup(Q5) ;
! e! V7 R0 s5 Q: h. `5 z/ @7 v: ZClassGroup(M);
6 H" b/ Y% O! l- F: A  {0 _ClassNumber(Q5) ;
- `# l: _; `% EClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
9 @4 M1 H9 E. f8 ~
QuadraticClassGroupTwoPart(Q5);
% l9 {8 M  r: q9 ~7 YQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

% u: \5 h  D. E* ?8 @9 ~; ^) yQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
( h( j1 Q: D4 Q* K8 DUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
2 g' j0 K# O% ~" gQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
# U) ]# D8 N8 `3 y! s; Z5 f9 GOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
2 [% d$ t$ p* z( o+ Ctrue Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
5 p: ^- K$ G+ O* Y    <w + Q5.1, 1>
]
-4
; `- f+ P1 B- d9 e$ s/ o
>> FundamentalUnit(Q5) ;
( v) j6 b. c: ]! O2 Y- y                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
" U& q2 u9 c$ ^3 T6 ?: o

& g' s! J# M, d0 _>> FundamentalUnit(M);
2 V6 a2 v! ]7 j$ q7 I3 I" I                  ^
$ ~. K+ i* g/ L. D/ U& K* BRuntime error in 'FundamentalUnit': Field must have positive discriminant
! T9 t! e! s: B: c& a4 g& a
4

9 |# }, B* s& e8 l9 b$ O4 w>> Name(M, -1);
+ H5 w! O2 y% t8 j( x       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
2 P4 A2 ^9 `: d( l& T
$ Y, x/ x. s; f3 w8 ]4 p1
. k  A& [. z9 i1 R; a, xAbelian Group of order 1
6 I# J: ~# o8 N. nMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
% Z1 \" `# t6 [/ L2 _/ o$ ~0 vMapping from: Abelian Group of order 1 to Set of ideals of M
  M7 d- D- o" L# W8 N( V- d' O1
2 q; T" E! `2 _& {8 P: X7 [3 q1
Abelian 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
( K) f- _% c8 m1 W& LMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
# _" o0 u3 w# p, }  n-4 given by a rule [no inverse]
Abelian Group of order 1
; i2 p* d0 `& i, NMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 G8 @6 x3 U; K& R-4 given by a rule [no inverse]
false
false
4 u/ o* F- ~9 o) c( R===============

Q5:=QuadraticField(-3) ;
Q5;
8 K, O" L7 ^& b1 V# b
5 h; m7 P* p9 n3 {Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
& U) A8 T* E6 T& k: k: w0 fM:=MaximalOrder(Q5) ;
" T5 Q# o6 `0 v+ G* v6 N: x/ X0 n) hM;
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;
: m- t2 [7 d6 a3 d( I/ _. ~: J" j$ IIsQuadratic(Q5);
0 k# Z9 S, d; f! ]$ v0 W3 Z2 lIsQuadratic(S1);
; f/ ]# g- M) ~& e2 _IsQuadratic(S4);
$ O* G) T- _! s# k& T- LIsQuadratic(S25);
$ Q7 x$ d2 O0 y; ]7 J4 u& kIsQuadratic(S625888888);
+ n! q+ N  U# x6 V, v& aFactorization(w^2+3);  
9 u6 ]; k- W1 B. x5 d' z$ @Discriminant(Q5) ;
, a' g2 s; W5 z. }) |: ?7 MFundamentalUnit(Q5) ;
0 J( J+ P4 V4 d) P: pFundamentalUnit(M);
8 p# R9 K; F8 f! R# eConductor(Q5) ;
% P3 S+ A/ P  V% s
! v$ a  o( {/ p* I! E' h9 sName(M, -3);
Conductor(M);
) J* B, O* D$ v2 l0 {ClassGroup(Q5) ;
# i' |4 Q: y, t8 P$ a( R. TClassGroup(M);
1 q8 U1 y& o1 M' S# ?ClassNumber(Q5) ;
3 i; s8 g/ v& s. F: bClassNumber(M) ;
6 J& |' Z) G. ~; APicardGroup(M) ;
; V. _8 U! J3 v0 s' n1 Q/ WPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
( ]8 y2 C. l: \, @0 a9 w! \: ^QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;
! N3 b$ r1 w1 F8 ]1 V! P
' {1 J0 o% T: [/ u- K# O5 ~: }8 ~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
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
3 f6 K% I  R: P0 u( C% x4 Wtrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
. S- {( p( u- [: y+ @true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
9 w" G8 m$ p, e8 ?1 N# R0 `8 C: d[
) ^& K' P) `7 P; O    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-3

5 U. e- f/ j+ \; V, |>> FundamentalUnit(Q5) ;
. F. K% ^" h  }" v; N8 R* _% Y                  ^
5 C9 C: b, z: D, ZRuntime error in 'FundamentalUnit': Field must have positive discriminant
. r9 v; L4 p" h0 ?

; v1 E" n) G% N* U6 B/ o' D>> FundamentalUnit(M);
                  ^
5 c  G- @2 h% S5 s" TRuntime error in 'FundamentalUnit': Field must have positive discriminant
% _/ o1 V! a) E& F7 q$ y
6 F  ^5 p7 _1 D. i: ^/ Y, Q$ l. i3

>> Name(M, -3);
( n. K: ]% Z8 x3 V       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
9 U& o. B0 m5 g% r8 g8 t: e- L
# r2 L6 i" d6 k3 d) K- O; L! v1
Abelian Group of order 1
3 v2 R  ~- K7 Y0 ^; qMapping from: Abelian Group of order 1 to Set of ideals of M
$ i8 `4 R$ i% ?6 K- vAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
8 n: X$ {; G( }! p6 M& J1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
# r# x: d5 @. v2 w& D/ Rinverse]
9 d, [1 [( l$ i$ v) ]; v! n1
" _$ L% U; c% g  b# X1 f9 v. @4 _Abelian Group of order 1
& Q8 P9 b- b6 ~Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
5 S4 L5 M; `6 P5 x& ^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]
# J8 N  A4 ~) C% h, \( qfalse
false

孤寂冷逍遥

lilianjie

Dirichlet character
Dirichlet class number formula

3 v: z' G$ r% 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 S' ?5 ]6 h. B5 P- i/ D! y+ G% T
-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

* ]4 `/ I8 \' D-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
4 k- W  C* j/ w& s; |
-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，

7 r! V0 r3 F$ w5 [$ E& u

h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

6 ~! ?  k6 M, o* q
% g3 G* }6 J2 M
-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

7 J& ?: i1 G8 e. i8 iF := QuadraticField(NextPrime(5));
8 V' {1 p2 {7 M( t  f+ d' c3 d
KK := QuadraticField(7);KK;
  E, l9 c$ g) G7 w3 \5 i4 ~( [K:=MaximalOrder(KK);
5 `* k4 w; v+ m: ^; D& ~Conductor(KK);
# B$ k+ R6 O. s2 l& xClassGroup(KK) ;
9 H7 e& Y: L: ^* K( X7 bQuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
/ l* B- I6 V2 l7 MA:=K!7;A;
  r) {- T' I4 ~6 aB:=K!14;B;
) {- ]9 L4 b  l% _1 r  }- S" s" jDiscriminant(KK)

$ \2 `. I! O6 o  ZQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
8 T6 _& w  q: T( s28
5 d2 a! r' d  a6 E1 N+ @Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
( I& r, @1 S  V  A; A. O2 WAbelian Group isomorphic to Z/2
Defined on 1 generator
  z: ^" M: c; f, n7 ~1 NRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
, i( ~) s) ^) T: ^Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
, I) p+ [6 n! Q8 X. z# }inverse]
false
4 Q8 k7 @0 x8 C. E( C7
" k* H. P" J/ Z0 o; ~) i( V14
6 x' Z1 e7 o( m( G( {; f28

lilianjie

4 v7 ?- B8 `- J2 z3 e
$ r8 c, V! ^& j( G6 o* v


7 a6 r4 X9 q$ p
. x6 n. u3 e+ R! H
1 v1 T8 O# _5 q( |! V
9 ^/ b& e* A; z* w5 n, \分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
5 e0 \1 Q9 ]; _% L9 x; rCyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
" F4 y  V9 q/ @$ W1 C9 L0 |MinimalField(CC!8) ;
6 G- W8 `& e: EMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
7 i4 f, H8 l/ X1 ]& M8 l4 \RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
CyclotomicOrder(CC) ;

& s. n! |4 I8 q: N  W$ }6 ]9 yCyclotomicAutomorphismGroup(CC) ;

9 t& M! J4 ?6 jCyclotomic Field of order 5 and degree 4
% @$ {5 f3 y/ q& Q$.1^4 + $.1^3 + $.1^2 + $.1 + 1
9 B$ j  c0 A/ A9 n  LCyclotomic Field of order 6 and degree 2
5 p/ i: R" D, N4 K, @6 E. j$.1^2 - $.1 + 1
8 V: l7 `  _/ V$ b( Q. ]Cyclotomic Field of order 7 and degree 6
$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
& B9 N. G% o  PRational Field
Rational Field
* N) I& a0 c$ _: D1 ^. `5 _* dzeta_11
zeta_111
123
7
" f1 L/ @. v2 a! x7
Permutation group acting on a set of cardinality 6
) J! u9 s0 p; Y6 _" z+ p  UOrder = 6 = 2 * 3
7 L: `- L3 q% ^+ c- S    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
- K% c6 V! V7 x* t& I" a- ^Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
  r1 [' r5 e' I4 v3 v. q0 sCC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
% w  _$ z7 p1 E7 rDegree 6, Order 2 * 3 and
5 b' k* x; v  K; [Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

* X% ~0 ?/ E) D* Y1 c8 ]

lilianjie 发表于 2012-1-9 20:44
2 H2 S8 d  S' N; B分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123

R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
, A1 C6 |) ~# ?5 e
6 v- ~' h' s  WQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
, ~4 w# [2 b, {$.1^4 + 1
2 z8 }  e9 n- G; rQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

# `+ u# {6 V4 k9 H(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;
& V2 s7 g! C( k8 vC:=CyclotomicField(6);C;
5 a3 n* d# a4 |FF:=CyclotomicPolynomial(6);FF;

5 D- E: X! T7 T+ ~F := QuadraticField(6);
$ n$ I, a7 j: JF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
  M# \6 f+ r+ q$.1^2 - $.1 + 1
2 t) }0 J: \3 L, h+ dQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
# v3 @/ M% c1 mF5 = factor(x^10 - 1)
$ Q  ]  d) Q. T/ ~5 b) t& SF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
" |  s% J7 x. a& h: L. i: ]1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

: C: M0 E/ a3 @; y$ a9 vQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

; g* b# t+ N, f' y2 e, s3 TF := QuadraticField(10);
F;
" h" P  w/ G8 {) L$ yD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
2 U7 ]; X4 W4 n1 l" aCyclotomic Field of order 10 and degree 4
: P, ?  V3 R/ w5 Q$.1^4 - $.1^3 + $.1^2 - $.1 + 1
$ K3 B& I0 V. D7 ^( }7 L) L4 k# I$ cQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
) m* x( C3 B; `# J9 N4 G[
" d* T* e  m* r) v# j& z+ t( a: l    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
- g$ L$ U$ Q+ ~$ x6 a& l]

lilianjie1

分圆域：123