实二次域(5/50)例2

lilianjie

& h& b+ r. n) w
Q5:=QuadraticField(5) ;
5 R0 @- d+ b% [Q5;
2 Y4 {2 t! `! r/ {& Z! OQ<w> :=PolynomialRing(Q5);Q;
6 f+ p$ C0 {1 o# `/ n4 H( O
4 E! g% D8 V' \0 V5 jEquationOrder(Q5);
* M3 J; s& _# z  sM:=MaximalOrder(Q5) ;
M;
NumberField(M);
% F# P$ i  Y" U- z) e  ^9 @9 YS1:=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: i/ e9 ]% T; V5 hIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
# {+ Z. _1 ?  W7 k, J$ l  lFactorization(w^2-3);
Discriminant(Q5) ;
3 t# n* u  V1 ~FundamentalUnit(Q5) ;
- o8 j" e! k+ d. l# I* `FundamentalUnit(M);
Conductor(Q5) ;
0 }6 A, j- r8 D3 m" g+ r8 HName(Q5, 1);
: o$ w! b7 n8 H% A% J* R$ D- cName(M, 1);
4 ?6 y7 `$ w+ v0 {' i2 t2 aConductor(M);
' d% D3 R$ c3 p2 a: c% C% H, A) F; sClassGroup(Q5) ;
# n  Y0 s9 ^! M" f8 _0 i! Q. c# w+ hClassGroup(M);
3 M7 C  V7 y$ ]5 p8 U- Q- FClassNumber(Q5) ;
ClassNumber(M) ;
1 b% W1 X8 P+ \
' u# k* z8 x1 C5 A" wPicardGroup(M) ;
/ ^* C4 [1 B6 S2 g( r6 E7 ~PicardNumber(M) ;
' b) D2 B; Q, R
4 ]" ^  u. {+ _. U
QuadraticClassGroupTwoPart(Q5);
; @9 a; Q: U- t- p) ]4 p2 |QuadraticClassGroupTwoPart(M);

3 Z+ `& L' W$ }& m$ h5 Y; ?/ V/ z
/ j+ s& f: g( D3 V6 {  v$ ?NormEquation(Q5, 5) ;
" ?# l* d4 c' r1 x- z: @  \NormEquation(M, 5) ;
) r6 G3 ~/ M+ d1 o5 h" a  U
  d* G0 U7 w  E; t( s3 x3 p
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
5 n8 @" z( [. o" m* d; g" VEquation Order of conductor 2 in Q5
Maximal Order of Q5
( M8 ^9 K7 j' r* M8 g$ X5 n' Y2 PQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
$ o2 o% \& W' ~) X' @* r* ?6 W1 xtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
; @% q& V9 m* z* j, @, ^8 z8 p$ wtrue Maximal Order of Q5
, q( ]6 I# T/ |4 J1 ]$ qtrue Order of conductor 16 in Q5
5 H8 L! c1 a2 S% ~0 r& p- Qtrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
- y% ?: l  ]* i5 F- A( i) _1 w    <w^2 - 3, 1>
- I1 o; k1 G8 _  G+ }" |! u8 S]
# R  J5 |" P8 y1 w4 _6 X  L  I5
; u) o3 G' z  h! Q6 S0 }9 g. E1 z1/2*(-Q5.1 + 1)
0 S  V$ d- y0 A& f, ~4 V-$.2 + 1
5
Q5.1
: H! a/ V! b! ^$.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
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
2 O5 M, P! d* Y$ {+ Q1 `6 QAbelian Group of order 1
) p6 F: f0 }0 _; @3 s/ ?1 l: }Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 I% l* A( u: T  @3 k% o5 given by a rule [no inverse]
Abelian Group of order 1
( O: `! c8 W2 G) kMapping 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) ]
true [ -2*$.2 + 1 ]

# z' x* O2 ~3 y# m$ E( D4 l
1 L7 E& L4 c+ w) t! i


% x$ x$ f  J( F( o


' i7 g+ E# g1 Q& ~4 V# N
6 w/ v. K* Z, W' z
9 w* s/ D+ Q* ]) l2 n
- }% _) n* F/ h' b( |  f; T7 k. K- a==============
. w8 A3 e: x$ x% F
Q5:=QuadraticField(50) ;
) g4 K- d+ O4 {" xQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
/ ]7 B7 j9 O' G8 gNumberField(M);
! z- m% x  R  B0 J2 {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;
/ B7 q4 l% r2 E( CIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
" G# b2 j# W0 v1 P' b* V* g; kIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2-50);  
0 P+ u) W0 J- T$ g4 YDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
! k& V8 m2 `8 N. FFundamentalUnit(M);
Conductor(Q5) ;

% X' s/ ?4 M0 \% FName(M, 50);
0 E0 ~& I  O% gConductor(M);
" X( z1 P  `* O: eClassGroup(Q5) ;
0 s* P  Y. |' J& G2 `7 S9 m9 @ClassGroup(M);
; y' _- g/ ~: T" uClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

) k$ R" r3 U5 |& C" YQuadraticClassGroupTwoPart(Q5);
+ J- K8 }' \- L, lQuadraticClassGroupTwoPart(M);
  \. y& ]' ]2 d  a6 NNormEquation(Q5, 50) ;
NormEquation(M, 50) ;

Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! w& Y/ K. U* ~Univariate Polynomial Ring in w over Q5
6 J3 G3 N" A2 V  `Equation Order of conductor 1 in Q5
$ I6 B' U; w" B/ I0 U) QMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
5 ?1 X, v+ v8 A% h; _true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
2 C! N/ y  {" E/ |2 t: I5 atrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
; f& @0 v+ m; j: N$ @% B[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
0 [2 S- H. t; L% e8 h! u# \; l8
  g1 r7 B: F% a8 @Q5.1 + 1
$.2 + 1
8

>> Name(M, 50);
7 F$ o! Q8 [: M  j       ^
  E) [" K) `4 ]( R! LRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

1
0 p* Z, l6 c- cAbelian Group of order 1
, D. N  R/ L5 _! p) K! ~' Q% {Mapping from: Abelian Group of order 1 to Set of ideals of M
- `. |* U9 y$ T; e0 s/ _1 x- Z) pAbelian Group of order 1
# G6 Q- D" K! ^$ r* V! u: X: \Mapping from: Abelian Group of order 1 to Set of ideals of M
1
+ v1 @: i' \, f0 m. `1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
, T; @$ e4 S# t' cAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
: E! |' K: U, P+ L8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
8 }" M, Q! ]! ?& U& x' r2 ]true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

. Y9 S9 q. E8 ?+ [3 R5 k
, w6 t$ N) J& ^, [( `4 M) Q0 R基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2
) ^# [2 `9 E- K0 T
, t( c+ N, {- b; [ x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
4 [$ n' D& ?& @& `& C6 p x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

( d4 |. W3 y$ w0 k
最小整解（±2,±1）                              最小整解（±7,±1）
9 T8 A8 c  S! r# j' k2 v8 q                                                             ±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

判别式计算Discriminant
  ^0 r6 b% p7 p. A7 V
0 A1 p/ P; G( l# P. Q- ~5MOD 4=1

（1+1）/2=1          （1-1）/2=0
7 u' ]! |" v* Y- K5 b: o
2 [6 G2 p! U1 z0 \D=5

& h; A" \5 L0 V$ g5 @
50MOD 4=2
. ^$ o# E0 @2 @$ H5 x+ QD=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：
. @, H% ]/ T' J( f# f3 T: VC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

* n( u4 X* O9 ^* {& W
9 ]. Q3 L2 I- [% ?+ W' _分圆域：
分圆域：123
5 p2 ~* Y1 s4 e
R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

. u, q3 u" g1 H: V) t(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
! K0 ^9 z' U9 i0 E
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
, c! V  i7 B% g* r& cFF:=CyclotomicPolynomial(8);FF;
/ B1 m3 N- J$ g4 R- k
F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
( u, B  f+ W% P( L3 d% [Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
* S! |, D/ u+ Y8 Y9 [Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 }/ Z6 ]) U% }: ?[
    <$.1^4 + 1, 1>
]
) K9 Z* w$ u: |/ r
R.<x> = QQ[]
6 e) A$ u, G  D9 R4 t- {8 _6 KF6 = factor(x^6 - 1)
F6
; x0 [$ R7 k1 H1 J1 l
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
$ g* C. a  m- l' m- O: _2 e
4 Q8 J2 M  u9 Y) t0 ?# XQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
/ C% B8 R  w( Q+ l, g: v$ g1 Q
9 w6 A& e/ S; [% q- D) y* d' P' ?F := QuadraticField(6);
- @& V" X5 A* F* ^F;
( P6 I. K0 r, {2 H0 n  O. k$ jD:=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
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
% d: i6 j) y0 \8 ?2 ~" h8 s& k, Y    <$.1^2 - $.1 + 1, 1>
4 Y! D( m6 C2 h]

7 o7 z/ e& ~# m$ Y; lR.<x> = QQ[]
, Z7 `- d/ G. j7 V2 B2 W/ ~F5 = factor(x^10 - 1)
4 Q7 I* H2 A  S' c8 KF5
3 B. w2 ^7 U* m6 F% f0 m(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
; N- `+ q4 r: p  n1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
; \; |% f- C! _0 d
: y5 C$ b% g" m- ?% q+ WQ<x> := QuadraticField(10);Q;
4 \" M! k0 _) n; a2 v1 O# i0 kC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;
7 N7 N/ S$ K/ K5 d
F := QuadraticField(10);
) e1 |% o4 |: @* N2 l% |/ \F;
D:=Factorization(FF) ;D;
5 Z9 I4 C  U0 nQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
. M" _& q! B% q' l4 k$.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 - $.1 + 1, 1>
: r' s+ h6 s$ A( @% l% Q1 T( L" q]