虚二次域例两（-5/50）

lilianjie

0 F' B! y* M# \! |Q5:=QuadraticField(-5) ;
% I( l0 ?/ }5 x( o7 Y  XQ5;
/ R6 u6 |' }3 S6 V) T
+ A: _+ u. n5 MQ<w> :=PolynomialRing(Q5);Q;
6 ^+ @) `; L+ A$ q, g7 w8 eEquationOrder(Q5);
2 T% |0 m/ a' s) _: ?+ t$ A, ]9 @8 EM:=MaximalOrder(Q5) ;
+ ?6 j" w  h1 ]0 e* j3 ]& M% {! o- uM;
NumberField(M);
8 [! b% v4 H# @5 Z+ i/ i6 N- ^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 k9 W2 r% Z6 N+ @2 ]IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
' q1 M0 R) z7 W, ~% `+ dIsQuadratic(S625888888);
Factorization(w^2+5);  
Discriminant(Q5) ;
* f6 \; ~3 g1 ]+ ]2 BFundamentalUnit(Q5) ;
& k/ G! Z) s! D+ d  d& F  d7 LFundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
/ \. U' s( T2 a$ nConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
! ]' B  o& e9 c  V. O' U- sClassNumber(Q5) ;
ClassNumber(M) ;
5 s3 F$ b$ L/ n  y; w6 Y/ YPicardGroup(M) ;
/ }; c/ \6 |& \0 F; e, @. ^4 n3 V5 B% tPicardNumber(M) ;

0 w% I( Y0 \3 ?1 O+ k6 f6 uQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
. R6 t" w- o$ g. L8 B6 |NormEquation(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
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
3 E! [. l; r+ ^2 n% w$ u" v6 b; ?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
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
4 {) J+ `& G0 u: u1 B[
- N4 S% D* ?2 l; g) _8 R    <w - Q5.1, 1>,
6 z  \. Y% M, l( H7 ~6 x2 b% Y    <w + Q5.1, 1>
]
-20
# L( C% F: z4 G9 V1 S3 X
' s! p- a6 T  T- s! N5 G  g' y>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
5 j+ d: z  n0 t4 k. b+ m, p1 ?

>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20
: m2 F! H! ]7 J" o1 J  W0 U
>> Name(M, -5);
       ^
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
$ a/ d% O, V2 S$ f1 KRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
% z! i+ g5 D+ j+ M' E8 i) Y* q    2*$.1 = 0 to Set of ideals of M
4 T8 j  y% ?: t" c( B. uAbelian Group isomorphic to Z/2
1 B' k6 K0 o4 z1 ~" h3 @Defined on 1 generator
Relations:
9 u! i2 Z: R( f0 A) S4 K9 j* v7 _    2*$.1 = 0
8 M, S4 s0 s% c; j- xMapping from: Abelian Group isomorphic to Z/2
* E6 I% n- H, WDefined on 1 generator
Relations:
0 {& U0 S# b2 u- x- Q" Q    2*$.1 = 0 to Set of ideals of M
2
& u% O0 k# q2 `9 E6 ^% d* N2
Abelian Group isomorphic to Z/2
* ~: U2 \3 i' S# K: y. lDefined on 1 generator
Relations:
    2*$.1 = 0
, A5 r, K+ I; O. n: ?2 iMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
: k7 D: S9 U7 J- f. c  ARelations:
( z( v8 c3 u" x) A    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
1 l( _9 e# R4 b0 I6 S2
Abelian Group isomorphic to Z/2
  y& s9 C2 S2 ^! ]3 ?Defined on 1 generator
Relations:
% d* v* \+ g. X) Y4 h1 d7 r    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
9 ]8 r9 J& F5 x, m8 c9 WDefined on 1 generator
Relations:
; Y- B# I; p7 o/ Z2 \+ H: X    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
: K6 a2 e) c0 a: x( }* o4 minverse]
" `( `6 R8 r% A0 v/ W2 G# kAbelian Group isomorphic to Z/2
. y4 Y. Y6 E+ k4 KDefined on 1 generator
Relations:
    2*$.1 = 0
4 p' \2 i* x2 {) jMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
3 ?% Q( L, Q+ f' y+ FRelations:
) M9 y+ {6 y. d7 z; J; ?+ M    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
  R3 l  u4 N& ?& ifalse
false
==============
  h: ^' b! t# |' S# D; V

Q5:=QuadraticField(-50) ;
Q5;
! D9 R: I5 z# X1 O) V6 B* o
Q<w> :=PolynomialRing(Q5);Q;
8 i2 H* a+ L' f8 ~# D# BEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
9 o& m  \1 S! E8 x/ {$ e& Y+ S* k7 JS1:=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);
9 R  R( N1 D! u4 N) bIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
& J3 @' E& _% y0 j# r: f* t2 `IsQuadratic(S625888888);
Factorization(w^2+50);  
' V# Q; U* q: q4 LDiscriminant(Q5) ;
8 s7 |* W) Y* N( b' {FundamentalUnit(Q5) ;
FundamentalUnit(M);
( h  }% O7 |/ C$ G1 ]Conductor(Q5) ;
5 _8 w% j( b, y9 l
' x# x' A" b+ _3 C8 [Name(M, -50);
- Q+ a$ f. H3 e2 b) mConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
( R% e/ S- Q+ \0 r( z) WPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
/ G9 L( @$ U8 g( Y0 B; a' j' gNormEquation(Q5, -50) ;
NormEquation(M, -50) ;
& n( U# L  R- f1 S/ `" J
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
6 t: w! H* p# d8 V2 G% TUnivariate Polynomial Ring in w over Q5
6 i9 I5 W" C, O% xEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
! j3 q: V9 j. h0 jQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
+ l7 K0 L% F: C5 iOrder of conductor 625888888 in Q5
9 _9 O4 ?7 ]1 y, U  g! Z# {3 \true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
2 [5 y5 t; h0 q' T/ k4 I. r5 ctrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
' F4 a( }( G% j% E. Strue Order of conductor 1 in Q5
  f  Z7 c& j" w' h/ ptrue Order of conductor 1 in Q5
[
. d( p2 J  H4 N& A5 |- s. h    <w - 5*Q5.1, 1>,
* A' o3 g3 g- ?" I; J% a) G    <w + 5*Q5.1, 1>
]
-8
$ X/ A! A' l" D6 F0 P9 D3 w2 I0 R
$ j5 e7 d0 q/ J  K. X* P( F>> FundamentalUnit(Q5) ;
3 w5 c, e! Q! m0 I                  ^
8 Q/ I& \! c3 Z& A& ZRuntime error in 'FundamentalUnit': Field must have positive discriminant
) n% O, G7 K/ r5 _7 A6 {) I

; v4 Q1 ?' H: e2 X) E/ r! D3 @4 t>> FundamentalUnit(M);
                  ^
  ]5 k9 e+ \' F- A1 j! Z% R- yRuntime error in 'FundamentalUnit': Field must have positive discriminant
7 i8 v, }' N4 }+ x' u
$ N: ~2 z3 F' _; i/ ~& Y8

* n6 k& L) t! q# r1 D7 F, I! C* l0 L>> Name(M, -50);
. H2 P6 V: T" o8 c4 L       ^
! g" u; H/ q6 f, E0 o' rRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

# n' d5 T  }. l% `1
$ C% A; c/ ^' g( Q' u) j" U. b9 [Abelian Group of order 1
. Q9 R+ b) s( p' v* j1 M+ o: jMapping from: Abelian Group of order 1 to Set of ideals of M
' R/ y7 F6 E: uAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
# V; N( o' J3 t: @0 m- T1
1 i( [8 ]$ m* @& r9 MAbelian Group of order 1
" `0 h7 p0 H/ |3 a7 MMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
) W: g& p+ r0 Q1 `+ d1
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
* Q2 _" N# I) v3 D- |% D& U3 x-8 given by a rule [no inverse]
6 x- a; O  a  H  H$ Z$ ofalse
- I0 u9 X" B8 B$ \false
; p* d; P6 Q5 ]/ a& y

lilianjie

看看-1.-3的两种：
+ c, y4 `: e/ m1 z5 d: o( H+ l
Q5:=QuadraticField(-1) ;
& j+ J  r) u$ [# p8 R. sQ5;

3 K7 b- t2 V# u4 z$ i9 lQ<w> :=PolynomialRing(Q5);Q;
4 R, B: \7 r9 x0 _+ REquationOrder(Q5);
( t  d1 U+ M% b0 b' T4 [) B4 rM:=MaximalOrder(Q5) ;
6 Y& B+ {2 X& `4 m5 Y" M7 b/ dM;
( j: X* t5 `/ z8 m, I$ Q# f# nNumberField(M);
* G$ V! n# S8 H) u- h. {" e9 WS1:=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);
Factorization(w^2+1);  
9 W4 a, v9 i5 F/ n* q, l. sDiscriminant(Q5) ;
" u0 F" r4 O' j( _FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

Name(M, -1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
* j+ o1 o! S2 p# h( ]8 V% tClassNumber(Q5) ;
3 _9 c2 }- \& \) u! B2 }ClassNumber(M) ;
1 g- S/ I' X" Q: }: C* a! b' }PicardGroup(M) ;
  d# Z$ E6 |$ u4 pPicardNumber(M) ;

$ c) T8 e7 x5 [+ c3 k2 ?" }8 ^$ ~QuadraticClassGroupTwoPart(Q5);
. X8 F9 s6 l6 [1 g. T) _1 s4 [QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;
- l7 N; u# [0 K* b1 [. |
8 y; e! Q! G$ x  j4 AQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
; X- u5 E5 T* n6 e/ R. h1 v& f" eEquation Order of conductor 1 in Q5
/ B9 [; ~/ V  t  `, J+ ]4 vMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
% b2 |/ ~! ]( y7 Htrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
6 |6 {" D2 N3 Y' H# u8 Ktrue Order of conductor 1 in Q5
  ]0 N; ~# D7 ?1 Itrue Order of conductor 1 in Q5
% M0 q8 B$ A1 q4 _& K* T[
    <w - Q5.1, 1>,
2 M. F+ h/ ^& s    <w + Q5.1, 1>
]
-4
6 k; `! B8 O, Z  b9 x$ t+ p
/ @3 c. ]' @& ?" M>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
. V* P$ G8 P4 c- @) Z; D$ L9 W
5 ^' w5 |2 M% O0 y
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

4
4 B0 @, I- o5 V$ E# a
/ y) g% c; U. }: C% g>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
3 \3 E# E- X. z7 O6 K7 e8 D  ~. t
/ @# E5 L  @3 @& A+ O2 ?1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
: r+ s% k) l8 z& MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
, R' G0 d& P+ ]! U0 E7 T1
/ v+ `! o* Q# ?- l- T4 u- w1
Abelian Group of order 1
: `+ o4 N& _/ V' Q% u$ qMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
8 h! o) u3 T. W2 Finverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
+ I* q" D4 E. o+ C* u-4 given by a rule [no inverse]
Abelian Group of order 1
5 _/ B- I4 |% o! B% dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
& L. A* N  X7 l, u) J$ t-4 given by a rule [no inverse]
false
) s0 q, F5 U' Z  _* _8 Xfalse
===============

Q5:=QuadraticField(-3) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
7 ~$ t4 U0 Y: v, Q: U- J. z# ]EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
& l/ J  Y9 q* F/ V2 P) M# dS1:=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);
$ J6 d6 m; W% iIsQuadratic(S25);
IsQuadratic(S625888888);
; I' U$ p  K7 s" PFactorization(w^2+3);  
Discriminant(Q5) ;
5 Q! P8 L2 Y  u8 Y& R4 r9 f: hFundamentalUnit(Q5) ;
# s/ n3 [. f* T* e2 G* RFundamentalUnit(M);
7 u% h4 Q+ d2 M4 I& ?# b) W; Y# [Conductor(Q5) ;

Name(M, -3);
( ]1 c* l$ @# M& {0 |, g3 iConductor(M);
ClassGroup(Q5) ;
' J( b7 ^) @' h# W. n3 i' N: KClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
0 E/ E6 N" G5 e; d0 wPicardGroup(M) ;
, p5 e2 _' S) s% o* R! d9 E- hPicardNumber(M) ;
  I) U  r$ I( b: h1 e
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
9 M7 s' {/ ], U6 R2 m) LNormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
( ]7 G! ]. h2 ?1 p$ aUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
3 Q1 t/ k/ T9 P' o) M: |: X1 VMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
& P; b. j. K- d0 Etrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
0 P3 N4 K  q: S) B$ X( ?6 Atrue Maximal Order of Q5
/ g3 d: M  b. }true Order of conductor 16 in Q5
2 G' Y) ]5 \) \8 F  f- V; w% d% btrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
8 t6 a2 F# F5 a[
( M. U( X0 z: O3 e& U9 X" g    <w - Q5.1, 1>,
    <w + Q5.1, 1>
* w0 l9 L9 X& w- c9 U- W1 a, T]
: h. g' u  }: }2 P-3
9 F# v5 C: c  {: B$ N8 A
>> FundamentalUnit(Q5) ;
                  ^
7 ]2 s, U$ M5 U& f: ARuntime error in 'FundamentalUnit': Field must have positive discriminant

0 T5 E* L% X( b$ p8 v7 o3 |8 ~
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
& G$ G. k2 |! L  M: x2 l5 i! _
3
. X$ G& z! I! [% W9 ?) ]& I* f" L# U
>> Name(M, -3);
) I* W. M3 d1 t; ?5 G       ^
% Y4 a2 t& M1 ]: {+ _9 n3 `% |% ?Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
3 a' B1 x9 L" x3 e
1
Abelian Group of order 1
% H2 n( f! h9 S: ], ?) }( fMapping 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
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
# o9 V& c% i2 t$ Rinverse]
1
! M+ c3 Q0 R+ D2 s! \( ^8 Z, K! ^Abelian Group of order 1
4 c! D$ {2 `8 M* q2 yMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 Q% T0 {. z3 ?* u  Y-3 given by a rule [no inverse]
! x& r7 H# y1 Q0 g" YAbelian Group of order 1
" j0 f/ U/ s- L$ C: L! _Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
( g6 K# w( H2 g6 g& j8 ?  m-3 given by a rule [no inverse]
' x: N% ^# p3 v9 L6 `false
1 C! C2 I1 j; w! B! ?; n) `false

孤寂冷逍遥

lilianjie

, D, ^7 W+ d* V% V* @
Dirichlet character
1 b: k5 u& H9 s9 E. N% YDirichlet class number formula
  S( r7 d; }4 o; g$ {
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
$ U: Z+ R; M: U2 c1 j# ?
-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

+ \: B+ I" g' B7 j' @0 v-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，

) Y% i$ z, r! O, M% [: e
. i7 W% j, v# L' a% {- H0 ]
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

; X$ k+ H  i/ w& d4 N( D

: j* M' u5 p8 ~: ]' X% `-50时  个单位根                          N=200
9 M) g4 F  L, l- x/ c

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

+ @8 u' ~4 m9 k9 `( H4 A6 {9 M3 J
; L5 M1 [& U. S' Q1 mF := QuadraticField(NextPrime(5));
' h3 a4 D8 e( v. g1 e5 ]
: F; A6 f0 F; R* RKK := QuadraticField(7);KK;
, m5 ]2 q, P! b1 J# YK:=MaximalOrder(KK);
. `+ z$ ^/ g( |+ t5 l7 ^( \Conductor(KK);
* t5 q4 z2 n! W3 J( h. JClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
' B+ f; \) T8 K# D* n7 s5 O6 r6 {A:=K!7;A;
B:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
2 V$ U; M0 n' W  Y5 `Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
' T2 a8 Q9 u! M% J  XDefined on 1 generator
Relations:
5 r, X3 N* g- b0 ~1 Q' ?6 S: [    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
* ?  Y% H% E+ m8 o! w7 ARelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
/ d$ W. ]6 z  Y$ ]false
/ H0 P6 F0 q' c% [* ]( @7 }7
, k9 J1 P) S, f6 m! D" E) e14
28

lilianjie

& h6 H0 W0 l1 c  o) ?
5 B$ a! M/ T# p# @% i7 o) F




; Y. z. J) o* R6 h. f
分圆域：
; g9 M7 B/ ^7 wC:=CyclotomicField(5);C;
4 n) m) Y2 j0 p/ `% j+ o3 R$ H9 V5 LCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
' }' b: U( q7 l- f% }3 {CC:=CyclotomicField(7);CC;
( p6 O/ t8 T& x1 `# LCyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
" F4 b1 c5 J3 vRootOfUnity(11);RootOfUnity(111);
) f# N/ B  M8 Y0 _0 |Minimise(CC!123);
1 O: F, y! F% n4 X. ~% [Conductor(CC) ;
% t1 x5 O( u5 M0 O3 o# rCyclotomicOrder(CC) ;
  q' _% T# z# H  H4 Y
CyclotomicAutomorphismGroup(CC) ;
: [) C5 b8 Z1 }
  o# T6 J+ s! h" j" t* j& sCyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
) j! t& ]$ c& XCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
( K' ?/ {2 }2 g- r# q1 WCyclotomic Field of order 7 and degree 6
1 \; g. l# s9 I" z$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
3 t( K8 o/ ]/ b/ o' Q, pRational Field
Rational Field
Rational Field
  e  P8 A- d) i( O" D" P3 L% c% Azeta_11
* ?& E' W) ]: g$ gzeta_111
123
. y" }5 E. y9 B( M; `( |7
7
9 Z( {5 s. j; `. C2 g: HPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
6 ^1 e' g1 @) c+ k5 f8 s4 q5 R( RMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
  b4 P* x; k6 s. G) H6 NCC
" c' y: n7 F4 C! {' N( bComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
# W2 d+ A% E, @4 s- w# ~Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
. \3 u; w" @& g  i8 |" MCC

lilianjie

lilianjie 发表于 2012-1-9 20:44
) f/ @' ~1 `4 W# @) Y分圆域：
C:=CyclotomicField(5);C;
/ G2 Y; |: M/ w6 Z, U5 \7 DCyclotomicPolynomial(5);

8 X; S% a  Z3 m
- h% i5 d8 t' @2 q分圆域：
. I9 X% Q! @9 g分圆域：123
, k" j# M5 t6 |
R.<x> = Q[]
- W" F, \8 X0 o1 UF8 = factor(x^8 - 1)
F8
* Q7 m: N3 Q/ w1 X0 e
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

* g' ^7 e0 l- Z# _% _, lQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
; r6 C* @+ v* b4 iFF:=CyclotomicPolynomial(8);FF;

* G. h. ^: {3 u# D, f( n3 jF := QuadraticField(8);
8 \- @/ H  N( `; cF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
7 F& f& |! b1 s. O$.1^4 + 1
  m8 A7 ^# ]; E& a/ z, U' {Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
9 Q* W+ \$ \2 l2 l5 s2 `- p[
    <$.1^4 + 1, 1>
) ?8 |+ E8 O1 H( J6 o2 R5 v]

  F0 x; n. M0 t/ ~R.<x> = QQ[]
F6 = factor(x^6 - 1)
, W( _; a/ d# j' H" [4 N( U- R; vF6
! _3 I" C% b" ^$ u2 D
9 H7 K- f  ]  b0 g$ n+ V(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+ o# h. [* b
Q<x> := QuadraticField(6);Q;
; C$ Z) O4 x5 RC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
% @  s1 o. A  a- ^. D0 u
& v5 a  g5 B; ^/ X0 q2 J! rF := QuadraticField(6);
F;
$ J; G( K  L3 n) L/ wD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
4 v+ R- `6 [- q$.1^2 - $.1 + 1
0 G* p  J) q  s. zQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]
; s, {8 y. i6 }
R.<x> = QQ[]
( Q" E: r6 }5 n" r. y7 A/ Q' L9 _0 hF5 = factor(x^10 - 1)
8 |: R2 a  n2 g2 x6 a3 l% yF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
7 g- L& X7 M) h! B1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

Q<x> := QuadraticField(10);Q;
3 N% V  k+ _& ]$ {8 Z- r5 O8 y, Y9 M8 iC:=CyclotomicField(10);C;
) E( k+ x  i7 q! |FF:=CyclotomicPolynomial(10);FF;
; p+ u) w5 n, n* }2 }  y2 @
7 Q( H- }' G/ n% |  c% bF := QuadraticField(10);
( ^7 o- I0 g" Q  ZF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
0 [% l. i0 ^! pCyclotomic Field of order 10 and degree 4
" j8 w' N! k0 Y# K) O$.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