实二次域(5/50)例2

lilianjie

+ {9 Y0 X/ T! C# x
6 o+ e! t, t; E+ u4 C1 `% H/ oQ5:=QuadraticField(5) ;
Q5;
; b( Q. d/ o; s6 K3 a3 M1 L( gQ<w> :=PolynomialRing(Q5);Q;
3 k0 W& B  H% k
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
, s' W; |; f; h* nNumberField(M);
4 i8 T5 G1 Z: ?+ ]1 J4 B& W8 qS1:=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^2-3);
Discriminant(Q5) ;
3 `( h% t! y  G3 R. D6 WFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
7 Z5 A  _; h, p# `3 F" R% iConductor(M);
: [# b* ~: x: ?0 N4 c6 P; G" x4 gClassGroup(Q5) ;
( r" q2 i% G$ G1 lClassGroup(M);
ClassNumber(Q5) ;
9 V1 @9 f$ C& `4 E; A" e: XClassNumber(M) ;

PicardGroup(M) ;
% `4 c3 @& P8 g4 TPicardNumber(M) ;
9 o+ t/ `1 w, q* l( p

$ @5 [. e) n4 dQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
( W7 O2 [, T" y+ x: x0 y

0 e0 }$ m3 b8 m# D* q& k/ WNormEquation(Q5, 5) ;
" D2 c2 x' k8 D1 {6 ~2 fNormEquation(M, 5) ;
1 e; b9 A7 z% {- M* F: L

Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
2 f1 v: C3 h' h: }( m! m  pUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
# ~7 @' }+ C$ H8 t7 s6 |Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
: I9 q. y# d$ J) T* e" Qtrue Order of conductor 16 in Q5
) _( K" l  s% g' |4 X# s) ttrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
$ y* q" p0 G" H0 @3 \! k5 j0 u* c& p[
) Q8 _5 h* V( R/ L$ x: m2 V    <w^2 - 3, 1>
& ?: ?9 J  [* ~4 b  ?. U3 _]
/ s9 u8 q, H7 t7 ?# C* _" E5
1 P5 a: d8 {( V$ i) F7 n+ W2 @1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
$.2
% E0 n- W4 X8 F/ F* _; v1
# i1 x' ^, T! b8 G! v0 CAbelian Group of order 1
Mapping 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
% ], L# Z; s2 q1
1
Abelian Group of order 1
3 w. b! @0 T0 \Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1 R1 g; ^; h; [0 V! I  @7 N1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
$ w  s! J5 Z8 O* I1 m7 @Abelian Group of order 1
& V7 w7 N" e& V5 b% F6 U( FMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
+ `; |' b, ?6 [( n# ktrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]


! H9 [( _& C& _
7 |/ ^5 J/ E5 @




( ]4 L8 g) b* A! |+ v: k$ }. c


! X1 d: M6 d* n# K+ @0 _& {==============

Q5:=QuadraticField(50) ;
  ?5 O6 C4 X7 E  o" ~' EQ5;
4 c5 G2 R7 n# R& F
% h9 R; t0 g/ K, @* Y( F# v: ZQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
# x/ Q- o5 y8 f$ @% \M:=MaximalOrder(Q5) ;
2 q& I9 ?5 O1 b" aM;
6 ?" H4 ^- }; {% 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);
- ^, F+ Y7 O( t, F+ X: kIsQuadratic(S1);
. L5 U+ X" C, gIsQuadratic(S4);
4 V; M. t; j& _+ S/ \( u0 e& {- qIsQuadratic(S25);
- q3 Z3 V( I  R* `8 L3 cIsQuadratic(S625888888);
Factorization(w^2-50);  
. a0 {1 V2 q' L3 g" P9 LDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
  Z3 n6 G- N, r# oFundamentalUnit(M);
" }& ~( v4 |: e) y6 z; z9 d* E3 nConductor(Q5) ;

Name(M, 50);
0 n$ f7 ^) `, [7 q7 [- KConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
1 }. L2 g- c$ Z9 d6 V; z5 Q  Z, E! F$ ?ClassNumber(Q5) ;
ClassNumber(M) ;
8 H! U* n, b( }5 C: {PicardGroup(M) ;
: h& @% Q1 r8 PPicardNumber(M) ;
  w" n& i% Q8 ?" s: q
! `) o% m, T: eQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
7 o5 p$ {3 h  C4 _& v3 RNormEquation(M, 50) ;

# F  H6 n4 b' A* @Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
+ [. Y- M& E, qUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
+ P0 V- M4 g# |" l5 Y" I4 d0 dQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
; g. e3 E' p  m! J* ~; i9 ~8 ^6 vtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
* H& y+ U( @9 \6 ^1 p0 _[
    <w - 5*Q5.1, 1>,
/ @) U- S9 v7 j8 `' f( y# k    <w + 5*Q5.1, 1>
]
! l* D: i2 _* S% `' l6 S5 A# B8
5 f, t; ]5 T& U5 }7 g& }8 [2 `Q5.1 + 1
8 ]. y! Y* Z7 r$ S" \$ l$.2 + 1
* Q* s* @: d7 o5 H3 K* Z. R) |8
5 F+ `3 Y% u5 G* k: V
; D' }1 L, I! h, k% h>> Name(M, 50);
       ^
0 ^, W( O1 P& h4 Y+ }6 bRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

7 K7 X7 F1 M% }1
Abelian Group of order 1
3 h8 E, o" F, P3 D# [Mapping from: Abelian Group of order 1 to Set of ideals of M
. |+ m) y$ M% X! u* GAbelian 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
( d" @# T+ X, H+ `7 Finverse]
1
/ q2 \/ a/ g& s0 O* I" ]! I% VAbelian Group of order 1
- {" S- a: d. R9 _0 rMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
4 @7 W5 g; E7 [5 g  e# mMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
/ {* }" A- Q% T# g1 v4 r4 P' Vtrue [ 5*Q5.1 + 10 ]
4 m& O& m' J/ ltrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

" f  }# `  B6 D
基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2

x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


最小整解（±2,±1）                              最小整解（±7,±1）
/ `, d2 f, v8 K( a8 S                                                             ±7 MOD2=1
. @0 \( Q$ T' \3 T3 D' u
两个基本单位：

lilianjie1

lilianjie 发表于 2012-1-4 18:31
7 R- E; C: C( I' ^, K基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

$ s$ t2 ^! ^. R基本单位fundamentalunit

lilianjie1

判别式计算Discriminant

5MOD 4=1
& l) P( e0 S$ W5 g0 @( I
（1+1）/2=1          （1-1）/2=0

2 A9 z* Z9 ?0 T( n8 C* AD=5
2 @4 v/ y5 R% p& I" L

50MOD 4=2
7 E) C6 L& a8 u7 w& Y# o* ]5 {D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
& ?. s8 l$ @7 n! L  @+ q; Z# Z! F1 }9 Y  s
, m( H8 J: k& y4 n/ Z; u0 b分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

. }$ m" s% t; E$ J
" X! y* |4 i% ^4 m分圆域：
7 N, f5 q- x! X/ o# i# J分圆域：123
/ T( ^/ l' _5 ?% |0 z
R.<x> = Q[]
F8 = factor(x^8 - 1)
F8
0 P# [+ a, l- k& x
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

Q<x> := QuadraticField(8);Q;
2 m9 ]) h6 g# P+ P8 BC:=CyclotomicField(8);C;
5 p2 h8 V( m. g0 h7 c" _FF:=CyclotomicPolynomial(8);FF;
* r2 c3 p) @8 |8 M: ?* `
& p& f7 k- f9 f4 k  R* o1 }) b; F# jF := QuadraticField(8);
F;
" g& A  E0 B  N4 nD:=Factorization(FF) ;D;
& ]' }! _/ `6 ~. w( _Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
. V* ^* @9 H' k* Z- A[
    <$.1^4 + 1, 1>
9 {2 C% I. r! M8 U; u; W; ?: s]

$ E0 S6 }' O& N  [# V8 `+ G4 TR.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

1 q. _& S: D  U# O(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
! D& U- @; y1 w
Q<x> := QuadraticField(6);Q;
# j7 m+ i  j% g. k& H5 B- zC:=CyclotomicField(6);C;
' Y: i' I& Z% `% }4 B/ C" IFF:=CyclotomicPolynomial(6);FF;

; e" H% K# `7 Y3 C7 W/ k; d8 tF := QuadraticField(6);
3 Z2 h. |: X8 ?1 i" L, F' ?F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
4 m8 ?1 e' g7 v. r5 v0 k, B0 ]' f$.1^2 - $.1 + 1
4 u0 P% Z8 O. }; yQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
( A. O# Q" w! G! o7 \    <$.1^2 - $.1 + 1, 1>
$ b# B4 `! `9 h$ j  x]

9 u* k/ l! p2 X4 |R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
: g- T! t! w0 Q# U(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
" G; H. N# F: J4 n& j, @& q- @1)(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;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

+ }. G2 q& Y+ `9 k! R) DF := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
. J& U, _) I( c2 A! UQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
1 F/ p. q; g; `/ \]