实二次域(5/50)例2

lilianjie

- G# H* q0 \" u. f' h( w
% F* \" k8 u  v7 b# `- ?" m; qQ5:=QuadraticField(5) ;
Q5;
( {% t. K( k. b1 b& N6 y9 `. r9 {Q<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
: Z: w; E: m2 ?2 `; h" FM;
; s+ N, L6 X; J3 }7 w& YNumberField(M);
" A! M, m8 _) W  ]* p' V. Z& d1 _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);
Factorization(w^2-3);
/ {8 Z- X7 x5 J- T' CDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
( i; B  L  }. oConductor(Q5) ;
; s6 m5 H3 V) rName(Q5, 1);
* i: c3 ?! I) Y) uName(M, 1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
- ^. U4 O1 b3 j% U8 WClassNumber(Q5) ;
ClassNumber(M) ;

PicardGroup(M) ;
/ u2 E, a0 l& d$ N: h& o5 f# U: KPicardNumber(M) ;


QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  A" ]: J* G/ `$ ]
: C+ ~  u% T- z+ {5 T7 ~
NormEquation(Q5, 5) ;
' A+ h- {' K0 y$ [' \NormEquation(M, 5) ;
5 U' f6 f: X& I7 r/ N
3 F3 |& V. P' Q# g0 `
Quadratic Field with defining polynomial $.1^2 - 5 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 - 5 over the Rational Field
Order of conductor 625888888 in Q5
6 I1 r- _! F: p( `% P2 W% I9 Ctrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
" B$ F* l& [( Otrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
, Q# x& `4 _! Z' x[
3 u3 V+ V; j6 |    <w^2 - 3, 1>
]
1 h, O# E0 }. p* S$ E5
' A2 A+ j' m/ B* M1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
+ ]9 c2 J7 T$ G* a$ OMapping from: Abelian Group of order 1 to Set of ideals of M
1
6 p' z" z" P6 f1
0 a. e- V3 I* ~" q% HAbelian Group of order 1
5 I. ?, l3 N/ I1 ^2 T5 N/ rMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
0 }$ q! n$ W1 \' l- @Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
+ m) c! C( R# P2 S5 Z6 s0 n+ Z4 V5 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
  p( h% g1 f/ _" v3 C5 W% q3 R5 given by a rule [no inverse]
+ d0 W* Y! b$ x) u. etrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]
2 r7 c0 \0 W+ r$ w
: e& t% f! Q! ]! d% r

7 W% [2 l  g. k2 Q% Q! a
4 C  }  [  c8 w9 k
6 C5 |# [3 V7 A7 I8 a! a" m( l
# _; x& {8 L3 f+ V! j. Q




==============

Q5:=QuadraticField(50) ;
Q5;
1 ~+ u9 x" O! z* q: L9 f
Q<w> :=PolynomialRing(Q5);Q;
( m$ F- U$ h0 u  w: V3 AEquationOrder(Q5);
% |( c8 F* t* X/ gM:=MaximalOrder(Q5) ;
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;
# ?  x, q; F5 tIsQuadratic(Q5);
IsQuadratic(S1);
: N* D- j5 H6 O/ y; N1 @8 [IsQuadratic(S4);
5 G. o! F# F8 ]IsQuadratic(S25);
3 e: ^& `" \3 F1 Q% \: `. QIsQuadratic(S625888888);
7 C* m* m2 B2 K& n- Z' M6 ]Factorization(w^2-50);  
2 |1 ]! w+ V4 q# @- D2 N( ~Discriminant(Q5) ;
0 L! ]. r" }3 U' g* W  rFundamentalUnit(Q5) ;
$ L9 W2 J- O( {9 E& E% {# {; A& JFundamentalUnit(M);
Conductor(Q5) ;

Name(M, 50);
! k- t* q$ y. [* c) T9 ]Conductor(M);
# ]; G! y! o2 t- M) L; VClassGroup(Q5) ;
ClassGroup(M);
: y$ D( J) Z5 N0 C1 }, R  qClassNumber(Q5) ;
4 A- l$ a- ^, k$ DClassNumber(M) ;
PicardGroup(M) ;
, M3 E1 a* g3 T: s( SPicardNumber(M) ;
9 Y% E  p: D+ C, ^
8 b' P  T* ~, q4 J2 H2 sQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
) C: D2 U4 A% y7 K: V4 _
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 ~9 @2 e1 c9 x. z8 y( pQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
7 _2 c+ T  ^4 [Order of conductor 625888888 in Q5
- C( ^0 u! G" {6 v) {* qtrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
3 J4 k. J7 @6 K+ e* D$ `$ C! btrue Order of conductor 1 in Q5
& S2 }9 d! y& Y( X" L[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
' P& {& u7 ?& F8
' M/ H3 D& G8 c! z" xQ5.1 + 1
9 E, @/ I8 \5 B3 t. k) ^0 X2 E$.2 + 1
$ O" L% I3 ?6 H! E; k$ C: _5 w8

- L7 x3 g. Q" x+ r/ Q# y  n>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

1
& F" }9 H1 v4 N; S; ~. J: rAbelian Group of order 1
" U5 h0 ~9 k7 e% ?2 g$ KMapping from: Abelian Group of order 1 to Set of ideals of M
5 S: O4 p! {) ^/ D0 K# A! L, vAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
% i0 Z. {2 [% XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1 g2 r8 ?0 ^6 t/ c1 C4 @1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! Z3 w' O; A  {& R$ @3 e8 given by a rule [no inverse]
( S4 {; B: i+ R$ b! H/ gAbelian Group of order 1
$ T1 I8 @  @: O; t3 UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
( v8 T7 a. _3 b1 N1 @' E# ~8 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

/ s$ v# {, a; T4 W# o$ i
基本单位计算fundamentalunit ：
1 [6 U. Y4 B" K0 F5 mod4 =1                                              50 mod 4=2

% _$ {. r/ b! |! z- d4 g! e x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
6 {/ p, E2 m7 v5 Q3 e% _3 U  e) V3 m8 I

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

) L1 D! m8 U1 f5 W4 B: {& Z& G
判别式计算Discriminant

5MOD 4=1

（1+1）/2=1          （1-1）/2=0

  ?  V9 I& t: k( Z- k% DD=5
8 E( i2 g6 m* g' h4 P/ _) @, D
1 |" D# e7 x) B) I
& I- v- r. s# V! B# U6 p: V& |50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

分圆多项式总是原多项式因子：
6 n" Y' G* v( M. h; M5 gC:=CyclotomicField(5);C;
  g+ g+ x: M: U5 ^& g7 K2 NCyclotomicPolynomial(5);

: [6 e( n. n8 E- ~  ^分圆域：
分圆域：123
9 Q2 I6 Y# c+ i& ~' M* l* O9 c
R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

. j+ ]0 c# d0 d- S. I(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

7 G  x6 C7 U* I& ^" ~Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
- B0 m, g) k9 n' KFF:=CyclotomicPolynomial(8);FF;
9 q7 P% c7 }* q- ]* {
9 h& g* |8 C  ZF := QuadraticField(8);
F;
8 H# k  l; [, M  s& bD:=Factorization(FF) ;D;
+ R# f2 e# I6 X+ O0 c# c7 M( _Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
  R) M' u1 L: [; W8 w$.1^4 + 1
' g) M2 b1 _$ m# @9 NQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
: @. |1 g+ N% u2 J    <$.1^4 + 1, 1>
1 i# C6 k" F: Z3 V; @6 I# S0 ]]
5 N7 l) V! l1 U
3 ~) V2 p! \3 H. @. U8 {R.<x> = QQ[]
( [& P/ k5 T9 E( ^. n0 t0 p& z+ fF6 = factor(x^6 - 1)
7 t% v! d: B( ]( R& b/ N. oF6

(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;
. h- B& @# w% O& C- _, R& sC:=CyclotomicField(6);C;
% V# D4 |/ U6 `1 f0 F2 v% n. A, pFF:=CyclotomicPolynomial(6);FF;
6 x# R7 \" p! a* O% r$ b
5 V0 {) q/ A1 `2 |F := QuadraticField(6);
' d( X0 t  z' v  R. l6 sF;
D:=Factorization(FF) ;D;
4 r1 Q3 m+ `6 t1 H2 tQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
9 l" m  ~- S  _7 R) p+ r[
    <$.1^2 - $.1 + 1, 1>
]
" k* i8 Q1 z' ], B& E2 i% n8 W
) Y* D- x& g( b. A' ]- d; `) ZR.<x> = QQ[]
% l# V* Q; Q, _& m7 nF5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
& t5 J8 |' o% N% e
Q<x> := QuadraticField(10);Q;
/ K" H) f% s  ^! l0 ?9 X0 f$ y/ m" I* UC:=CyclotomicField(10);C;
7 w" a2 L1 p4 {: g/ T* `2 {' vFF:=CyclotomicPolynomial(10);FF;
# Q1 s5 j3 N% m, {) i& w2 z, h6 `0 Z
F := QuadraticField(10);
  M( n, }% G9 p' w/ a  X, @  _% rF;
+ |  I& ^8 I, n3 Z5 PD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
" X  @2 U7 o9 U8 eCyclotomic 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>
]