实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
9 l' N# u8 Q  h$ RQ5;
Q<w> :=PolynomialRing(Q5);Q;

, K) B( a) \' t2 H  Y! J# I& D/ xEquationOrder(Q5);
' P% x2 l, {) P# Y: ^M:=MaximalOrder(Q5) ;
M;
% F$ B" n& F. INumberField(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);
$ v( W$ l# n; a4 z+ WFactorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
" G* c( B# y1 VFundamentalUnit(M);
Conductor(Q5) ;
0 {( W3 o) K# [& r# P- D+ s1 gName(Q5, 1);
Name(M, 1);
6 Y+ \: J8 J( [+ L% ZConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
1 x5 M* W; J! s- L2 w  w) R' b* S1 fClassNumber(M) ;
: Y0 j4 ]! n$ l# ~: O/ j
PicardGroup(M) ;
: x( @: u6 t2 Z$ ~2 l. }PicardNumber(M) ;

& t; d  T: K, L6 D: v* {
* B) T7 x+ G6 ^+ z% |# JQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
8 ?* {9 Y4 a! Y+ C8 v
: K1 u0 F% z& u7 W* v5 o1 t- L
1 @' ^+ P  T. [( Z) m9 `NormEquation(Q5, 5) ;
  P9 Q  K+ V( K% ]2 q5 gNormEquation(M, 5) ;
, p7 c- ]- T: y. R

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
( B7 A2 l  K5 [3 u$ d; _- X; k' A- NEquation Order of conductor 2 in Q5
5 T4 _: L. y* i# C4 j& T+ F* O. NMaximal Order of Q5
' K2 v) K0 r, T# c) _4 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
true Maximal Order of Q5
" ~& A) a8 v9 P- o7 I9 q( D& ttrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
9 q1 ?2 i5 M! @1 l5 J    <w^2 - 3, 1>
$ c" k1 M5 x4 E# `+ o& _# U]
5
1/2*(-Q5.1 + 1)
-$.2 + 1
  q; ~4 _% s' K- I% H( V5
Q5.1
4 z6 p' v+ \. T( F) x: s/ y$.2
3 d* F: y  c; P% r1
+ |% ?# V2 n* I1 k; r: u, r) iAbelian Group of order 1
" A, [) m2 l4 x# L8 }3 k1 _/ c; K8 \Mapping from: Abelian Group of order 1 to Set of ideals of M
: H7 a, d: g+ [Abelian Group of order 1
4 M3 h9 i2 B1 L6 ^; r; {2 E% }Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
3 Z- s9 P, [- O6 YAbelian Group of order 1
7 \! S' T* u0 C0 E" P1 z1 O6 D5 s# b2 @Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
& h$ d8 C+ ]6 T0 P, s/ {inverse]
$ e. n6 a$ P  w: I" }; {1
+ }: X9 R2 M) u0 {  d- ^Abelian Group of order 1
$ P8 M- t( r5 a; t3 R# |9 z/ U2 C- UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. X" m% A! a4 q* R& L2 s5 given by a rule [no inverse]
Abelian Group of order 1
- |0 w1 `7 e4 ^# ]$ SMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
# T% T9 G9 h( k* k  v5 t8 F7 strue [ -2*$.2 + 1 ]
# I4 p+ y7 i$ `- K7 q$ v

9 R2 x! y, s& \2 J
8 ^. O% e; h8 r& s/ ^' g



3 r+ r# c9 ^, q( K6 {1 L1 U7 `3 c: t



" }  u3 w7 N* h; b- B==============
) [; i8 C/ y% n) G, c6 ?
Q5:=QuadraticField(50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
+ V9 A2 |1 `9 y# H- U  lM;
0 u6 n2 ?$ w2 l- dNumberField(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);
0 i, `1 o: D" Q* A# n' K/ u' G( RIsQuadratic(S1);
IsQuadratic(S4);
0 S9 G5 Q& \) I6 C+ YIsQuadratic(S25);
% i. g5 t. ?4 u4 `0 X' z: iIsQuadratic(S625888888);
Factorization(w^2-50);  
( f8 J3 k* |; G* W1 WDiscriminant(Q5) ;
! L' R. l# X- z* HFundamentalUnit(Q5) ;
FundamentalUnit(M);
! T5 B  _( Q  {) z+ V8 KConductor(Q5) ;

( t9 d$ W) t% L3 V0 X& gName(M, 50);
& [8 m7 V; z5 S, b5 wConductor(M);
ClassGroup(Q5) ;
$ J$ \  {. ^& wClassGroup(M);
% r/ D/ u5 g. _9 MClassNumber(Q5) ;
ClassNumber(M) ;
2 k" z9 h/ T, _PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
  P& H$ c4 C; E) L4 PQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
% ~7 w0 d& ~, NNormEquation(M, 50) ;
+ E1 j0 j% R: q
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
4 V0 @$ |# T; XMaximal Equation Order of Q5
  j9 q, I/ O. J- D, }7 EQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
( f4 X" [4 J0 {& b" }# ~3 uOrder of conductor 625888888 in Q5
: o: X( c. T) u2 Q5 q, J& n2 b5 R$ Jtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
+ u: l9 z3 ]2 S/ i. D$ i$ atrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
" w' l6 j( g* x0 ^- ?true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
  V7 d7 Q( |* @7 i* V[
. G0 Z, i" F7 R! X) j    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
8
Q5.1 + 1
: P5 y$ f6 a' L' k; o0 h$.2 + 1
' B! y" Y5 v# R  Q- J1 K5 G8

8 `6 h8 V( S% [6 o) d$ _>> Name(M, 50);
       ^
: W- B; {8 J" y; k( tRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

: \& t/ K2 }; M1
' T* p6 W8 k; L8 ?# [% \+ M$ N  \, zAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
, y, A% R: R" l* y4 IAbelian Group of order 1
1 r4 M( [2 a' G8 ^Mapping from: Abelian Group of order 1 to Set of ideals of M
: d1 J$ @- g5 I' p* C, P9 ]1
9 k# y* v  E. C7 H" j1
9 N# L9 v" s# P4 i' q3 nAbelian Group of order 1
! D; C3 w* Z$ d5 ]  G/ v5 `4 [0 pMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
$ e. W2 _$ R& H1 Xinverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
$ g5 ^5 @# D7 ?( c' }/ D$ }8 given by a rule [no inverse]
) v% a$ d& t- N7 p* ?+ W' bAbelian Group of order 1
3 t( t' p2 c' BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; a' x5 U) g- e5 C3 |! h0 f8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

. x! W1 l% g1 s/ Y  ]- g. C+ G基本单位计算fundamentalunit ：
& x. M5 _9 [+ V$ x5 x5 mod4 =1                                              50 mod 4=2
- G! _6 ]/ e7 G- r/ N( W* s9 [" p0 q
0 h/ Q( N4 M  T  K x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
' H2 p, G7 R) f$ B$ ~# Z. K x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
( m: i) `0 }4 v( G1 ?# P

# x1 |) b; n: O1 a7 \6 [- O8 p5 A1 u最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1

两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

$ E/ ]# W/ ?* f; Y" {8 [' x6 T
. d% l9 E' o# U: ]0 t9 z判别式计算Discriminant

0 S3 Z% H- j# y- c# Y) @6 c0 H5MOD 4=1
; _) ~9 N8 l+ ]  b! s
& t1 i1 l, N! w" ^, E（1+1）/2=1          （1-1）/2=0
9 Q* e0 L1 e0 B
D=5


50MOD 4=2
& Z0 K( m9 R/ rD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
+ ^$ G5 P) `$ ]9 o% C& e& C6 g! T+ A1 Y
: Y/ e8 X! T+ s( X% \7 C分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

分圆域：
: q1 W) \! u$ c, L' }' D8 B7 J分圆域：123
( r/ q9 ]& ?8 {0 g$ {1 S
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)

4 T( [$ a6 c; q. l/ i3 ^Q<x> := QuadraticField(8);Q;
. A0 f" u- c' P9 c* AC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

" G9 Y9 s6 s* w( mF := QuadraticField(8);
4 v& M. y9 y, QF;
1 m* n1 n) \5 b+ v& a2 QD:=Factorization(FF) ;D;
- }' O/ z1 }5 o3 \/ }5 G2 kQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
6 m2 x. p3 E3 [- X8 j$.1^4 + 1
& w# f6 X, q7 G/ \Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# J# z' t- p, _* [- O2 d: n/ @9 F4 `! L[
1 O; A8 \& x1 ^1 w    <$.1^4 + 1, 1>
]

8 h7 n' M5 v4 j+ k1 Q7 T( e) {, \R.<x> = QQ[]
F6 = factor(x^6 - 1)
F6
" |% w0 ?2 r5 G# B9 v/ A" E( y2 H" X
: F! k3 N& R7 Q# j4 z(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
7 w  ~+ n: I+ W+ c( u
" N; z: w! Q- K8 b, v( wQ<x> := QuadraticField(6);Q;
8 T8 |7 v# R1 ?+ xC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
. E- W2 |  o; O" {9 B& a- QF;
D:=Factorization(FF) ;D;
- ~" [5 t  l2 ~5 \2 H6 tQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
3 S4 t6 w7 N# C3 ECyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
1 o" A! V; A! O, ^Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
7 N# Q0 G7 `/ [0 |6 M4 A) i]
0 A2 B/ |4 a1 ]7 _: \
3 K. `! c1 c* O4 E1 |4 N0 h; SR.<x> = QQ[]
! q  S$ Y  h3 m, {5 n! z) RF5 = factor(x^10 - 1)
5 c+ J# M! E; h$ H& VF5
(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)
% v' b( B6 z$ {- D( C$ h; R
( |, k1 U% [! w$ A2 W- xQ<x> := QuadraticField(10);Q;
/ g: G- ~' E5 S2 j$ WC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

, [5 w, k1 j) v0 uF := QuadraticField(10);
0 h# H! g. o6 V( X5 ~F;
' i6 q; `+ y5 W$ X  F8 L0 {" eD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
3 E! Y3 d* e8 T2 x* CCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
. l$ T& A( }7 Y[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
  B- J  W* i8 [8 k9 H) h' n. a]