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标题: 虚二次域例两（-5/50） [打印本页]

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作者: lilianjie    时间: 2012-1-4 17:41
标题: 虚二次域例两（-5/50）


Q5:=QuadraticField(-5) ;
4 F# a. B/ \# D0 Q  ?Q5;
- a+ _. [3 r6 N0 `3 T/ ~
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
/ V1 r3 l" U- m$ ~$ z+ {M;
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;
& s# ]: ?( u$ p, V! _IsQuadratic(Q5);
* f; C6 i" m* v" E' R1 dIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
: c' A/ z' m5 S5 [: _* g$ y8 @IsQuadratic(S625888888);
+ N2 ~6 F6 |' Z+ ]6 ~Factorization(w^2+5);  
+ u2 u2 g! K9 t) G# IDiscriminant(Q5) ;
& q" X8 e9 H- L1 q8 hFundamentalUnit(Q5) ;
! ]% |/ X7 h: M5 X- w* [) g8 }7 r' TFundamentalUnit(M);
, f, Y. G% u% ?Conductor(Q5) ;

Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
/ x/ E, N( M* j+ SClassNumber(Q5) ;
* P2 K2 D+ }3 k" p2 p, y8 w0 kClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
8 _* m1 r# z' h2 t% p4 X$ l
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
8 j2 d* Z  W1 M4 m1 cUnivariate Polynomial Ring in w over Q5
3 R- y. ?' l' S7 i  pEquation Order of conductor 1 in Q5
( G8 b& ~, ~* r/ h; R5 w, O4 k! yMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
8 \( Q4 E$ i' P- C$ z1 v" Jtrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
+ S1 m. @: {* k7 N$ [( F0 P, ^+ ^true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
# Q9 I) Z7 K7 ?8 o$ V[
9 Y! {# c9 k% f! p( B3 O1 o    <w - Q5.1, 1>,
$ p( M* j" I, {9 w5 H, B# [    <w + Q5.1, 1>
) C, `' l# t3 I7 G+ F  a1 X! x]
-20
. L3 Q) f* A  Q
5 y3 X. ]# O7 Z' L9 z>> FundamentalUnit(Q5) ;
                  ^
, Z8 r* f2 x3 K8 n- J: Y* l$ K1 _Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
3 Z9 B7 b: z' b+ r) O8 Q3 l! r                  ^
; |6 m/ p$ H& f" C9 n5 lRuntime error in 'FundamentalUnit': Field must have positive discriminant
9 _$ x- O! d0 K' e0 c
20

, G" _" P; Q) j. q>> Name(M, -5);
4 Z& i4 A6 t* \3 k       ^
% n1 [0 r: G- O. u) M- z/ q/ b) dRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

, B( |4 |/ S; k0 Y) A# `1 C% ^1
7 d! Q/ c* A; V7 `, v7 H- kAbelian Group isomorphic to Z/2
Defined on 1 generator
5 ]1 @- A3 w! X7 G0 PRelations:
  }2 M2 ?2 X6 t    2*$.1 = 0
  B! {( c5 M+ t8 M  @8 _" d5 IMapping from: Abelian Group isomorphic to Z/2
5 v4 G4 }* n: c" ^4 p4 KDefined on 1 generator
' ~8 I& G( ?* e  `& u' Z# IRelations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
+ @- v  b; i" J0 o5 I6 q# [: F8 RDefined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
1 C' L* |+ W9 D6 NDefined on 1 generator
Relations:
# f1 u7 D$ _$ _- U1 q  _" C- @* \; b, k    2*$.1 = 0 to Set of ideals of M
2
! O% \; y, y! ]( H& S( q6 D2
* U% F& E' p9 o  \( w; ?Abelian Group isomorphic to Z/2
8 `3 y& L& R# i3 fDefined on 1 generator
Relations:
/ P0 |& B+ ~5 X# b- F$ U    2*$.1 = 0
$ r$ ~' A+ k$ qMapping from: Abelian Group isomorphic to Z/2
' W# ^1 P3 F0 h! M5 MDefined on 1 generator
' l3 Q) }3 T' n) MRelations:
$ d: y5 O% U; {4 ]  J: }( S+ \    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
& c! }8 A6 }& V& z1 F/ Q4 V6 [Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
  m! [" t' i' N0 C* nRelations:
& T6 f, l* s5 r9 x7 g6 m9 g    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
5 {' i. P! x* ~* hAbelian Group isomorphic to Z/2
Defined on 1 generator
. J) v5 u# @! ]6 {) M6 A7 z0 QRelations:
8 v, v0 @6 x  H  K    2*$.1 = 0
% c. v9 o  M; c* ~8 A1 RMapping from: Abelian Group isomorphic to Z/2
# `+ @2 c+ }& A, m$ JDefined on 1 generator
# C& F; p, m1 f) v- NRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
, ~: f- C  Y$ T2 X% h& Tfalse
( E6 w, u" ?, z2 P1 u& rfalse
==============
/ u  M! \. W2 M' D; U; V) ^7 i3 K  l

) M0 {2 E0 }$ F+ K* }/ j, ZQ5:=QuadraticField(-50) ;
% j; h# A/ F. {1 b* u8 HQ5;

Q<w> :=PolynomialRing(Q5);Q;
/ t" S3 M+ K- P& }+ JEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
+ _1 F7 X7 ^) H8 D# f4 u; [: KNumberField(M);
4 P1 n" H! v) [$ L% C  R) 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;
IsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
& ~; z" F  c- e+ D. K& E! ]2 gFactorization(w^2+50);  
Discriminant(Q5) ;
8 E: v, D5 U8 z- k1 p  yFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

+ x9 }: G$ I: }Name(M, -50);
' x. f! o: O. G# C7 R6 I+ a* z7 p- nConductor(M);
! R6 Q9 R* s  y3 z) H- h  \ClassGroup(Q5) ;
& K6 A( V" o+ j, UClassGroup(M);
/ N% ?, g' W' E0 P# @+ ZClassNumber(Q5) ;
# K, _$ x! N( l- |# U( E( QClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
! _. M; T! R8 J  i
0 g( \# j/ w5 \8 kQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
3 K. R5 h$ y# |0 [- rNormEquation(Q5, -50) ;
' h$ Z5 l( G9 ~9 ANormEquation(M, -50) ;
) Y4 ^8 Z) x) k, c8 Q* I1 b
' x1 B2 _4 C1 h9 f  IQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
' O/ ^5 \$ K6 i5 L/ k$ \Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
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
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
/ J; j! R0 Y: s4 x5 l1 b[
$ B- O/ P( u- c    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8

9 K+ y3 |5 m, Y+ h+ C6 I>> FundamentalUnit(Q5) ;
                  ^
: M5 P" T! k3 _# g; t9 rRuntime error in 'FundamentalUnit': Field must have positive discriminant


0 m. U% k) m7 R) O>> FundamentalUnit(M);
) r) t, o" ~. J: X" e                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

; [, w8 f2 D& {9 d7 }2 Q( ^9 I8

>> Name(M, -50);
0 t6 b! Y  b9 j' b0 j       ^
, ^2 r, \4 I4 U: p. d) M0 CRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

1
7 K' g) e' W$ j$ _; M5 n- z0 PAbelian Group of order 1
* z1 _3 {5 ]8 b- ]" vMapping 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
8 c8 u- V. E, j0 v$ ~$ [1
1
Abelian Group of order 1
. Q  S: V. ~: T" f0 R2 FMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
" I, M7 g0 `) ^2 ~inverse]
1
Abelian Group of order 1
- D* R& `. J. }! g3 a. n+ ?Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
2 Y$ |3 y2 G! \( U/ A$ U! C-8 given by a rule [no inverse]
. k2 O. T/ s1 n4 p; _% i" BAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 w4 u- o; ^8 o0 L# g3 @, Q" O$ G-8 given by a rule [no inverse]
false
false

---

作者: lilianjie    时间: 2012-1-4 17:51
看看-1.-3的两种：
" h9 f$ I! |* K$ {
Q5:=QuadraticField(-1) ;
7 j% u. Q/ \  r( _5 \" FQ5;
* g" _  z" A* u' A! \, b
7 G1 S* {8 t6 v. x) a6 u+ ^" VQ<w> :=PolynomialRing(Q5);Q;
- y4 W' B/ O% JEquationOrder(Q5);
6 F3 ~" a7 A& A4 |M:=MaximalOrder(Q5) ;
/ i5 f/ O* a; t" F3 dM;
& Q! e9 W" D' T# S* J! z& ENumberField(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);
: B5 H7 ^9 [3 x/ z3 k* gIsQuadratic(S4);
7 z5 N% t3 x- ?IsQuadratic(S25);
# Y' i4 J- r: U  y/ HIsQuadratic(S625888888);
+ ]  q& a) d2 A& ?1 n% m5 VFactorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
3 v6 S) _- [- c" x/ o9 D  WFundamentalUnit(M);
* H# x' i4 F* S$ u% b% Q3 zConductor(Q5) ;

9 k: ^. ]# Z" ?  k$ n) FName(M, -1);
# E+ ?( ^/ u: M3 |Conductor(M);
( j5 Z3 n6 ^3 eClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
3 ~  k9 D+ f2 _& gClassNumber(M) ;
2 s8 u1 q7 C+ V. J: O" c- ~, ?PicardGroup(M) ;
+ ~- e' `6 H- Z  p' RPicardNumber(M) ;
: {& N5 A+ l6 C8 h4 F0 b# d3 s  e: O
QuadraticClassGroupTwoPart(Q5);
- B- _8 w# h! \7 a$ F  ^QuadraticClassGroupTwoPart(M);
9 R& K8 J. G" A% v$ P, h# X7 `NormEquation(Q5, -1) ;
NormEquation(M, -1) ;
5 q8 \5 r+ @3 E# v
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
+ Y) v. H7 g8 D( w- T3 iUnivariate Polynomial Ring in w over Q5
* e6 @/ o' m: f) C: vEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
! y" h. U, o, V) m( ^' wOrder of conductor 625888888 in Q5
  O9 D  B2 H* Z; v$ J$ F) ctrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
; M+ V0 x8 i  h2 _$ @  E  ttrue Maximal Equation Order of Q5
* S+ E- }# n% u1 C; Gtrue Order of conductor 1 in Q5
: Q5 e2 J4 ]  n+ f/ Ltrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
1 i0 v  ]( c7 s0 g) ]. N[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
5 z3 E' |( {, N, `]
-4
. ?6 g. o) K9 e1 w  I
>> FundamentalUnit(Q5) ;
                  ^
1 t( B6 R! T" \" s$ f( _Runtime error in 'FundamentalUnit': Field must have positive discriminant
" u, }( X1 n' t0 e9 Y' [3 k
, q5 a5 N7 H- u+ }4 r4 n
6 @+ o! j/ |* A/ I$ Y>> FundamentalUnit(M);
: C, y+ H0 F; v/ a1 r                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
2 ~% y; W6 `6 Z  R  n& N  `
4

>> Name(M, -1);
5 l+ z, P1 \" E: Y9 }       ^
. u/ k/ K* v' V/ {2 I+ I/ Q/ |1 E0 {3 dRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
6 M! H. |  m- {' y( Z2 B
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
: H" S8 I" k7 P. z7 U4 `, P; d2 OAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
9 S9 P( v9 y9 \; |6 N1
8 z# Q, i  t: ]' F9 E# S, x  a1
Abelian Group of order 1
0 B+ n9 O# E; q* HMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
% B$ x9 O5 l5 z* R1
9 g9 J; a! i- W8 l9 _- LAbelian Group of order 1
Mapping 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
-4 given by a rule [no inverse]
false
false
( X3 E" r$ l% w===============

7 ~% A: y) [( @, G8 G9 y$ p. xQ5:=QuadraticField(-3) ;
Q5;

( ~. r0 L4 P# }2 L( pQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
7 @' x0 }0 k4 b+ EM:=MaximalOrder(Q5) ;
5 E/ f3 h% E& V* _: [6 E3 i5 |M;
1 m  w7 `( E7 h$ T+ PNumberField(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;
+ S  p! {0 j+ g- @7 AIsQuadratic(Q5);
IsQuadratic(S1);
" m6 j% `0 j+ Y9 C; WIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
: s2 s1 V$ \9 }Factorization(w^2+3);  
8 U3 d& s2 n: w) [" qDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

, }6 k! `7 \/ c  p3 {Name(M, -3);
Conductor(M);
ClassGroup(Q5) ;
* V. S% x- b3 E, ZClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
7 s! D  w( T5 d  IPicardGroup(M) ;
PicardNumber(M) ;

. a- `( V, I" q3 E5 Y' @5 x' YQuadraticClassGroupTwoPart(Q5);
. j: U' f0 ?/ m9 `6 \9 LQuadraticClassGroupTwoPart(M);
6 S. u7 H3 {* H, oNormEquation(Q5, -3) ;
1 y& Z3 ~0 l3 ]8 {% XNormEquation(M, -3) ;

1 B2 q% A# K, _9 F  F& F/ ~Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
$ t. ~6 r- S2 d7 d/ DUnivariate Polynomial Ring in w over Q5
: i. Y0 b+ Z9 v  HEquation Order of conductor 2 in Q5
Maximal Order of Q5
& \2 d$ U1 `& ~5 l6 q3 X( B2 Y  @Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
+ k$ D1 n$ @8 |0 G8 W0 u3 @% |2 }Order of conductor 625888888 in Q5
- V  k* A7 ?3 k+ v+ ytrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
: Q; a8 ?4 S  i0 q' r0 b( U- ~true Maximal Order of Q5
" A! ]' T( j& |6 ?9 qtrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
) m4 u0 L  t3 s; {# O, Vtrue Order of conductor 391736900121876544 in Q5
# @! U7 \' |1 ^9 T5 F) q3 H  V7 y% O[
! S. d/ {. T8 W  M8 ]    <w - Q5.1, 1>,
) G( z0 C  Q: N, y( z0 S    <w + Q5.1, 1>
+ o; {* M7 F! c8 W, r]
; L0 [% ]% }: f$ E% \! {8 Y' i-3

>> FundamentalUnit(Q5) ;
                  ^
! W- P- ]: Z" g3 B# _Runtime error in 'FundamentalUnit': Field must have positive discriminant

3 K; v9 p; Y2 G
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

3
4 S8 s$ j/ U* S- W6 ?, p! o
' V6 V6 n- t8 x' L>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
; _8 U/ b. X, |7 N
1
) P+ k0 Z! w9 f5 Q7 VAbelian Group of order 1
2 ~5 w( p1 d/ Z- R. l3 e. cMapping from: Abelian Group of order 1 to Set of ideals of M
4 h! R. G* s8 m5 yAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
. {. f( N) d; K( g$ F) C1
9 K0 M/ e$ u! Q/ A& T& qAbelian Group of order 1
/ b3 G% W* w6 J! u6 R& D8 AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
' B9 k5 m9 t6 k" S$ hinverse]
1
5 J$ Y! Z/ ?0 J8 t5 a3 F3 sAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 x& r+ V" O) [: M-3 given by a rule [no inverse]
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]
false
3 a5 e) t2 L1 W7 sfalse

---

作者: 孤寂冷逍遥    时间: 2012-1-5 08:36

---

作者: lilianjie    时间: 2012-1-5 13:02


) k5 G& W5 h* }2 H+ y1 qDirichlet character
$ K. b' |2 z  X  G# |: ADirichlet class number formula
0 j# ?3 L: `* T. i4 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
  O6 m& e1 ]- ~6 `" W$ i! A
-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

: P5 y' w% i/ L4 D7 e-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
6 s5 j8 ]( l0 F! c) X- zh=-6/（2*3）*Σ[1*1+（2*（-1）]=1
! u& v' [+ e  r2 [5 L
4 ^% O3 [# N6 K0 Z/ m-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，
* r9 @# g1 P. b5 o& I) Y/ E# m

$ c7 h) ^- M6 O
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2


7 i) K7 C) s& m( K
-50时  个单位根                          N=200
( @6 t4 ^! k# Q

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作者: lilianjie1    时间: 2012-1-5 13:51
Dirichlet character

21.JPG (79.18 KB, 下载次数: 241)

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作者: lilianjie1    时间: 2012-1-5 20:37
pell   equation

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作者: lilianjie    时间: 2012-1-9 20:28

$ B; \+ Q" W7 ?& L
% q# _- ], D7 f8 |* A0 x" K1 UF := QuadraticField(NextPrime(5));
# H! s" ]( l4 r8 v9 O! f
KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
4 ~$ W0 Q' Q$ ^) f( W( zClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
4 Q9 B4 f4 b4 y/ m6 aB:=K!14;B;
Discriminant(KK)
2 _) f& D9 V/ O  \  z
7 b$ `- z3 g1 k+ a  h# NQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
  i# X! w" S% v1 g3 W5 MMapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
# A# B/ \9 w' r! bRelations:
% e5 [( i7 D& y2 @4 z5 @: X    2*$.1 = 0
5 j; D8 H& y5 a& x% C7 E+ z2 FMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
! k$ h& D: w6 T- ~Relations:
* O0 L' h( ]+ k" r4 L3 O    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
/ x, x5 z4 k$ D- [inverse]
, b) w9 o+ H* b$ {  rfalse
6 H; r9 s/ X- @3 u$ D( u2 H7
1 M) b% m1 a$ F( r4 D  N14
& X5 E! j! B5 c+ w7 a; P28

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作者: lilianjie    时间: 2012-1-9 20:44


$ r. v' b* \+ ^: l* P; i7 N

! }/ Q6 |& q% ^& V' y7 G


* {" l6 m& e8 E" F. A% L- X
9 ?. V; s9 w- d分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
! q8 N1 ]& D5 L- C' P; R' a1 X6 ~! kC:=CyclotomicField(6);C;
1 \/ V1 F! M  `) YCyclotomicPolynomial(6);
4 P( ~2 s$ C* ICC:=CyclotomicField(7);CC;
. z; D4 t: ^- ^' ~$ B% ECyclotomicPolynomial(7);
, T6 t3 k* L* N2 _6 ~MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
' |6 N6 _2 l( I. j! l+ PMinimalCyclotomicField(CC!7) ;
, U# [/ L" G! \# M1 RRootOfUnity(11);RootOfUnity(111);
$ h) N- I; g* g+ K! Z: @9 EMinimise(CC!123);
- x* R9 L% D! x7 y# N, Z) wConductor(CC) ;
  D" c% Y" @1 ~7 q2 X7 p8 G% hCyclotomicOrder(CC) ;
0 n7 P% O3 c' |9 c' r; L
CyclotomicAutomorphismGroup(CC) ;
" [2 h- _) i" L- _' n5 ?# m
Cyclotomic Field of order 5 and degree 4
$.1^4 + $.1^3 + $.1^2 + $.1 + 1
  E9 Z- J! ?. A0 A2 N( _! JCyclotomic Field of order 6 and degree 2
2 ^( F+ v, G, S$ b+ @  v4 j; }$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
. u' Q( I8 L. J1 e$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
# E! }( u, C# A2 N* A) ARational Field
# k+ x3 N/ ^$ }# xRational Field
Rational Field
/ W6 J$ q) f3 iRational Field
zeta_11
zeta_111
$ i+ b9 l0 b. _' D9 g123
1 k. r8 S. l+ ?1 V$ o" U5 Q7
7
! _5 ~" P! f* [* a' L. d- A- PPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
4 p. X) {2 M. x" M2 j    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
* Y  @' J2 C8 M+ s+ }) N9 O1 EMapping 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: $,
Degree 6, Order 2 * 3 and
! f$ Z9 u; n1 w. q# WMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
& F( B6 ~5 G7 e0 pCC

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作者: lilianjie    时间: 2012-1-10 11:34

lilianjie 发表于 2012-1-9 20:44
# |& i$ x- Q5 ~+ |* L( Z) g- y分圆域：
: g* A( U1 k' w" eC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
分圆域：123
7 c/ s+ g2 i# @/ K
R.<x> = Q[]
F8 = factor(x^8 - 1)
F8
  L- k7 Q# T6 ^' M) z
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
  e! N% x; u; B( n' k
Q<x> := QuadraticField(8);Q;
3 D+ W6 \- A1 k& \- \4 T5 U% L3 tC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

7 B  w$ g1 E' _& G' EF := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
  E; V  [7 B; fQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
2 m5 ^* |* U- F: e9 ]7 d1 ]0 jCyclotomic Field of order 8 and degree 4
3 ~' {/ d) m+ z2 ~+ {4 q- U$.1^4 + 1
% A  V1 e: h6 Q! zQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
0 T* P- \9 ~) I% S$ I  P, j[
    <$.1^4 + 1, 1>
+ v& ^( t; g4 |$ E6 v* _5 U) Y]
& h9 w2 V3 {2 l$ d! W$ e
R.<x> = QQ[]
F6 = factor(x^6 - 1)
  j  g6 R0 c$ f4 |; D7 YF6

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

; [' T0 D. ~6 A  `Q<x> := QuadraticField(6);Q;
6 k2 M7 A. B- `: @C:=CyclotomicField(6);C;
4 z2 V' M6 d: L& u6 P3 i9 ~FF:=CyclotomicPolynomial(6);FF;

8 t: v" L; v" G+ w3 N% Z# s- jF := QuadraticField(6);
4 ^: j& \4 b; M/ F( J! }+ ^2 jF;
D:=Factorization(FF) ;D;
7 {- g6 L$ P1 eQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
" C3 v2 Q( L; y4 S  n" O. t. z9 ~Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
6 t2 s' x4 H9 e! N; v; m' `[
    <$.1^2 - $.1 + 1, 1>
]
7 n* e* g) w+ Y2 x- M5 D. d& f4 y
R.<x> = QQ[]
: n9 w  K/ w: p( R/ l2 Q# `F5 = factor(x^10 - 1)
: ~8 E3 P$ N  S8 F* E" I4 RF5
: L+ M0 C0 u8 Z: [, s+ S' g(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)
( _9 ]' \$ o3 }1 i8 c
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
, a) G4 _( k" L- b$ b
F := QuadraticField(10);
% I& ?8 k5 l6 G1 k4 ^F;
  a2 m; O; o+ b. P2 R" ND:=Factorization(FF) ;D;
6 j, G8 i4 I+ }Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
( _5 L/ ^$ d& W1 U- A3 L0 HCyclotomic Field of order 10 and degree 4
' @9 v$ p7 Y2 D# Y2 J& B$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
2 y# U1 u% C( S6 P; N[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

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作者: lilianjie1    时间: 2012-1-10 11:55
分圆域：123

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