实二次域(5/50)例2

lilianjie

& G3 f* ~/ X! R6 P5 N. {
Q5:=QuadraticField(5) ;
! b; V$ U/ Z+ Y* FQ5;
' C8 P# f  m2 P- `1 E: rQ<w> :=PolynomialRing(Q5);Q;
& c, u0 g8 x7 I2 Y& y) O& }
EquationOrder(Q5);
M:=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;
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
9 ], }/ n/ ]' f" ?. jFactorization(w^2-3);
Discriminant(Q5) ;
- e' A, |6 Q) f- W( c/ o6 b/ XFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
% T4 j$ G* F! s' W" `Name(Q5, 1);
Name(M, 1);
Conductor(M);
ClassGroup(Q5) ;
* \0 Y$ g4 v) S$ EClassGroup(M);
ClassNumber(Q5) ;
/ s( A, R; c# W# U; j4 v; t* \: [ClassNumber(M) ;

0 ^+ ^: R" `! D# ]9 m4 jPicardGroup(M) ;
PicardNumber(M) ;
  _5 [- r. z6 d
* B$ w5 [2 y7 h. D  a1 \& W8 o( w
QuadraticClassGroupTwoPart(Q5);
+ G, G/ X* C9 T7 lQuadraticClassGroupTwoPart(M);


NormEquation(Q5, 5) ;
NormEquation(M, 5) ;
$ Q2 F8 ~$ P: a5 q4 P
7 b; }0 O* ?! t* @7 J/ ~
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
* B: n, ^6 G+ y' J4 u  @5 ?, ftrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
4 @' S- s% N& N9 E: X- c& utrue Maximal Order of Q5
true Order of conductor 16 in Q5
' ]$ R3 z6 ~- p6 e  n7 b% E( @true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
0 K; ]' K5 R$ s0 g; G, B2 [    <w^2 - 3, 1>
3 B6 N9 @( ~8 I. |# Y3 z]
5
1/2*(-Q5.1 + 1)
-$.2 + 1
3 s. t+ [  V, U' X2 q+ K: f# d( x1 T5
4 W+ \# p7 d" aQ5.1
$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
$ j. m4 [" n# g0 w. K# e! wAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
8 D/ C" S; t3 b: m& ^1
1
: m4 }5 X( p. k: V5 OAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
/ a3 F0 j, @3 H" T' Q3 d7 R. a5 T1 Cinverse]
1
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]
+ A. j& u$ u+ W" q8 u1 vAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. ^% _1 P8 ?$ _0 Z" e, [2 s; j0 V5 given by a rule [no inverse]
! @' t5 x. P/ h& K# l" P8 itrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]

9 D, i0 U3 a/ [' A2 F9 k- U


4 o6 t. l. a4 X6 Z8 }' M# X; G

, t. V, Y* @# F! v
3 q' d1 W# Y  D+ B3 J

# K" w4 i; i8 G# ?0 S) U! b

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

+ [: Y% ?% P1 PQ5:=QuadraticField(50) ;
7 m5 p% O# ?0 E! CQ5;

; m6 A/ w; _& S1 u1 jQ<w> :=PolynomialRing(Q5);Q;
7 x4 p! V. d) K0 ~EquationOrder(Q5);
7 e1 v; ^, W  j1 _. rM:=MaximalOrder(Q5) ;
0 x* _7 |+ `2 rM;
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;
7 c+ B; B$ f5 t) [IsQuadratic(Q5);
6 F2 {$ C) w: T; ]# t# c- ^IsQuadratic(S1);
8 c5 b! b, o! P2 h" QIsQuadratic(S4);
! b% `% @' M) _5 g3 @6 U; vIsQuadratic(S25);
IsQuadratic(S625888888);
/ T4 D7 ~' q0 M, c& n: w, `Factorization(w^2-50);  
Discriminant(Q5) ;
6 w! m7 k+ K2 n$ a; pFundamentalUnit(Q5) ;
' n9 C" b+ ?2 [/ e# R# e' mFundamentalUnit(M);
Conductor(Q5) ;

8 k+ ]  S" n8 H! ^. `Name(M, 50);
0 ]) B+ }% ]7 c+ v) N2 e4 C" Y- cConductor(M);
ClassGroup(Q5) ;
' s; ^& c: [+ UClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
' x! \  k$ W" @PicardGroup(M) ;
PicardNumber(M) ;
1 a5 C, l4 Y4 F& O: v  ^: }+ g
0 ]. J, U! F+ \  p; S, O2 eQuadraticClassGroupTwoPart(Q5);
- I4 a4 d! @# ~3 R0 z8 Z) HQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
5 Z: T. i7 r, |3 T$ `. c/ O. ^NormEquation(M, 50) ;
+ y& `, s* L+ K
/ C4 s! Q" [7 I$ g% _Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
: i# r  N; m+ z! kUnivariate Polynomial Ring in w over Q5
3 H' Q& ]1 f) k9 D/ ^' e% e# cEquation Order of conductor 1 in Q5
$ F0 G5 ]/ t" \* i' Q( m9 YMaximal Equation Order of Q5
' L" c" j$ T8 ~6 U# ~Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
* F! G  M) \$ n3 H1 t" zOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! s1 Q4 R+ `9 n: ?2 btrue Maximal Equation Order of Q5
& _: j9 q6 A3 ^3 Atrue Order of conductor 1 in Q5
, l) w% d0 Z( _: ^" jtrue Order of conductor 1 in Q5
6 e, k$ y# @: G  T! s  Ttrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
  ~$ _: @4 t2 }- @1 o! T8 i" b]
$ g5 G+ Y$ n5 W8
Q5.1 + 1
$.2 + 1
8
' |/ x$ R- a; ]9 S
$ E% F# q; }% F  g# S5 c>> Name(M, 50);
: R' ^% K4 C# ]: f       ^
% y2 U/ O% k3 J$ L" [Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
  p# p1 G: u; p; ]- V7 N  u
9 k7 n% _+ y: f8 o0 W# ]1
; ~: N& _% [/ o/ I) u! iAbelian Group of order 1
9 }) {" Q: ~5 m% @8 @8 N8 @; wMapping 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
9 H0 ?: C% i8 A* f8 K1
9 g& l' Z& _. j8 ^3 M1 _1
) |: v+ p$ x- M9 T( ~Abelian Group of order 1
4 b5 f* B9 P2 Y; I' K9 A) S+ I5 aMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
0 F5 J3 r% T6 linverse]
1
2 O4 h* Y+ c- V7 }* E* lAbelian Group of order 1
9 C- T5 j" i9 oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; s& F! A/ U5 K$ o% ^- F/ j8 given by a rule [no inverse]
( ?2 ~! K- k# n9 C! GAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
/ m5 J) ]7 Z) ^0 _5 o! y/ Htrue [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

! r% q, @5 n+ [9 A3 q  X2 z
基本单位计算fundamentalunit ：
- H7 I, W* f, R! L6 n; k5 mod4 =1                                              50 mod 4=2

0 y" i# |+ E7 B3 F, V9 y x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


& w( R& m- v; a1 W% ?/ N4 E8 }最小整解（±2,±1）                              最小整解（±7,±1）
. m5 `; S/ I8 p' O. J# j                                                             ±7 MOD2=1
# A8 a& y7 f% T+ Y& t
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

. |2 Q0 \4 O- a
6 }/ w9 a- b& @3 O+ B3 N% C判别式计算Discriminant

- d0 d+ c8 E# F* W6 H5 B' l5MOD 4=1

（1+1）/2=1          （1-1）/2=0

& l' W$ l3 r- a' R& T7 B8 xD=5
5 {7 Z. ~5 W* p, w2 ~
* v% v/ x+ k8 J  w5 M" q. C: Y
50MOD 4=2
0 s  t) _" i' g3 F4 T  b+ s- vD=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：
# E- k' b1 @% O" ^" `C:=CyclotomicField(5);C;
$ ], X- n# }# \3 X5 n6 i9 a9 LCyclotomicPolynomial(5);

分圆域：
& M- U7 ^. Q5 [% R0 ]分圆域：123
* c/ u7 c( N* f% p5 F$ b
7 V- I- G& P5 `0 g% t# o- h4 A/ wR.<x> = Q[]
F8 = factor(x^8 - 1)
2 ~. ]7 ^$ _6 ]% n" XF8

" k9 u; Q* x/ M. Q(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;
1 k' ~$ Z6 P/ y# u3 H, K2 @C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

# H1 c. S1 S, t" U3 X. E) R% QF := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
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
: E" x6 D  U3 T8 }9 I[
$ }8 B. c% o1 L9 R* B7 `    <$.1^4 + 1, 1>
4 _5 W5 ~4 i0 o  I]
) c1 V  f+ T: b7 z( b4 O4 E/ f9 A9 i
R.<x> = QQ[]
: M- @$ K3 p. r) m- [F6 = factor(x^6 - 1)
. _# B7 m  P, MF6

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

6 r/ e3 y+ h* b3 Q( }, v: ~# qQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

6 L' a( J" u5 \F := QuadraticField(6);
  S+ U6 Q  ~' Z9 L) R- AF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
. x9 z4 T- @- x' }# N0 i& S2 G! hQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
0 |# O: y  o1 N  }[
    <$.1^2 - $.1 + 1, 1>
8 m8 D- ]5 W2 V  ]5 @7 b* Z]
* x5 y: B# H* C
R.<x> = QQ[]
, ~* a9 H) Y5 ]: ?F5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
3 X# }% t8 P% L1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
8 C9 Q. C/ D+ _* {3 p
Q<x> := QuadraticField(10);Q;
9 s" L# ]* \" |4 D+ {" UC:=CyclotomicField(10);C;
/ O& I* ^3 J: T% `# sFF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
3 ]7 R5 V$ N8 Q3 yCyclotomic Field of order 10 and degree 4
. E; |  f6 L( E8 e% t. S$.1^4 - $.1^3 + $.1^2 - $.1 + 1
0 ?* u8 n' O' q6 J' ^8 [Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
% m3 x. Q  Z1 p; E/ H[
3 I, F8 A$ g8 ~7 s- ~7 ]8 ]& i    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
, O! }7 W+ ]/ }1 x8 D# Y! @]