实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;
, g9 q2 t) r5 i0 x0 Y( h- W5 P2 L
% d! ]6 H3 f& t& oEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
$ g  n- I$ r! M. b( t. G" }M;
NumberField(M);
1 s1 d- x: C: M# {$ wS1:=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);
4 j8 y+ _! C0 SFactorization(w^2-3);
Discriminant(Q5) ;
( Q9 a+ ^( R4 C8 HFundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
Name(Q5, 1);
Name(M, 1);
Conductor(M);
' y% R3 k& C; M$ s9 Y( N8 k9 F' ?ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;

PicardGroup(M) ;
% B8 e( T$ T4 \! x( T* C5 ]5 M! fPicardNumber(M) ;
7 k/ b9 w! Y* \- N
! Q5 w! e% e9 W. `
3 ^" g8 `% v9 M# w; z/ X4 Y( T' EQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
* d, v8 F2 P' R& o/ A7 \
" h- u) ?8 G; o, _2 o
NormEquation(Q5, 5) ;
4 e' J8 x0 A% z- iNormEquation(M, 5) ;
1 H: n) u- y4 U1 z, I% Y

: ~. `8 `3 T! C  ^/ L% `. {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
5 J5 y3 Z4 V! C8 d; U9 C& RMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
0 b+ u8 i# S% e( @; Xtrue Order of conductor 16 in Q5
1 M/ u# ]& a5 c$ c" e: wtrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
8 f0 g8 N+ ?% J+ W[
* v4 |% x& y! G- ]4 f  U  {* \    <w^2 - 3, 1>
+ M7 F3 H7 Z6 W4 w5 L8 r( S; ~/ f]
4 S: `* i# Z3 {. ?3 U5
1/2*(-Q5.1 + 1)
% r+ j' R' |7 U9 S9 Y. ^! D3 J% e+ P1 s-$.2 + 1
5
  I  H6 y, a6 o( h% Y( R# t2 \# H8 KQ5.1
$.2
0 a% l& \! w6 ?. p' D& d- t1
Abelian Group of order 1
+ o: |- ]; D3 O+ x8 V& KMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
9 L% t. T* \) R) M8 \Mapping from: Abelian Group of order 1 to Set of ideals of M
1
% a5 F3 H: w4 D1 B/ K1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
1 i! S, m. Y7 d7 o2 m" Ainverse]
1
Abelian Group of order 1
5 `' ]' D# w% P2 H" l$ n% c$ r# o* xMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
: t" v: Q' S  @  Q% W' Y5 given by a rule [no inverse]
Abelian Group of order 1
9 T( o7 K% |* V; LMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
: |' d/ A, c' @, {/ o+ @+ {9 r; U5 given by a rule [no inverse]
: k8 k" w2 m4 C. s" ztrue [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]

  }, b9 n2 p$ z! i1 e* v1 K6 F

8 ~% Y# R1 U' l0 `! q; }
% }6 W$ o1 U. y4 ^: o# |6 M1 G( G
; l5 \6 l5 D4 O3 [5 @( j* d
1 g+ W3 p+ O4 z, N3 `: G


# s9 N5 O+ v: L- b. D6 r1 r" K

, F& {, z' h$ o% @. Y/ X1 @==============
! R& I5 F: ~: |! Y4 ~7 |: X3 O* W% ?
Q5:=QuadraticField(50) ;
. g0 ]4 a1 i: b( C5 Y. ], `Q5;

& k5 X- O! a# {Q<w> :=PolynomialRing(Q5);Q;
1 P5 i, D+ g/ a# U$ X$ r7 s* oEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
) |) g2 A( K# I0 AM;
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);
# Z5 v: u- i) l" `1 C6 hIsQuadratic(S1);
* ^+ q  D% l; k. bIsQuadratic(S4);
( T) o! {; @; {3 u! j' SIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2-50);  
' K' n( j  m. N" i" `Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
7 U9 l4 m7 l# w3 K
Name(M, 50);
0 _5 P" a; z% ^/ M. O8 ]9 lConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
9 F. h- R: `2 x- |5 FPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
NormEquation(M, 50) ;
* E/ d* _4 x6 V" c% o2 w
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 p9 h6 P- c! Y$ {* qUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% w" f) d4 j$ d" i0 j9 h8 kOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
+ N  C2 o* T, h6 mtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
! W8 B9 o3 @# }" Otrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
  L2 k! X  A& M    <w - 5*Q5.1, 1>,
6 |- e7 J- [; D8 Y& W' a# P    <w + 5*Q5.1, 1>
]
8
Q5.1 + 1
1 o) @+ |3 g5 W$ G1 i' A& E0 U$.2 + 1
8 Q1 F. P. C# V; e8

& e3 y1 I- _, k4 I( X7 k/ s. @0 U>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]

% A* u$ s6 l. X; v% M1
; U8 B& U( g* p$ ZAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1 I3 e% h) b1 D8 Q) tAbelian Group of order 1
8 c/ {9 q1 G; A6 bMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
5 C) o* }: U3 j% G3 N# @' K' |4 e, gMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
3 W. I" {  n8 K" {7 y% ^" IMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* I: v( S4 `. j5 g9 A: e8 given by a rule [no inverse]
; @3 F' I8 o+ z( i5 s" htrue [ 5*Q5.1 + 10 ]
# T2 E$ y5 M. ]# htrue [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

基本单位计算fundamentalunit ：
7 b( }" e! [: R5 mod4 =1                                              50 mod 4=2

$ r0 i- G3 Y$ _) L! S+ p, A x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

2 O  W7 i8 w( \! y# C% N
最小整解（±2,±1）                              最小整解（±7,±1）
                                                             ±7 MOD2=1

: |: @  r! p( w: A2 }. o4 {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

5MOD 4=1

$ m0 F" z9 {+ d6 C4 g! F（1+1）/2=1          （1-1）/2=0
% ?. ?0 H$ z! S2 Q8 l
D=5


) j: Y$ Z$ Q4 ]% k50MOD 4=2
D=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
) X$ k% s! U. l2 c" U# K
分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

  @- M' a8 z4 u' B. r2 d1 v4 d  R
分圆域：
& n- @3 K6 D8 T0 `分圆域：123
( t% H6 d; j* `
0 i/ f7 Z2 B/ y+ P8 LR.<x> = Q[]
+ s) w. T$ L, U- Y! w, oF8 = factor(x^8 - 1)
F8
' |" w1 b: Y4 ?% ?& w$ }8 W
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
6 p4 e% W5 T! b; D- v$ c! d- o
2 ^* w+ M( l' i2 z% g; S: sQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
; t% O2 I. G9 ?2 ~- kFF:=CyclotomicPolynomial(8);FF;

6 I5 g9 G$ K3 w" P* wF := QuadraticField(8);
F;
) }" A* S9 b$ C7 c  yD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
1 v' ?$ [# `5 ]6 Q0 A0 OCyclotomic Field of order 8 and degree 4
6 R: t  [. _, e7 x4 G+ C% b$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
. X. Q. _- E# ]$ h    <$.1^4 + 1, 1>
]

R.<x> = QQ[]
, m1 _  }2 a) g! Q* qF6 = factor(x^6 - 1)
F6
$ E4 i* {! g+ c5 ?, V
(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- `, P$ D9 d* M: x/ _! Y: X2 S# A# @
  v2 d% Q6 `* W; CQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
0 S- Q8 c% T4 G" O: ~  ^' xFF:=CyclotomicPolynomial(6);FF;
; w; b. N4 {# s4 l, a) w
F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
- U. e# A2 C! o* V9 T* y- v% WQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
8 [9 X- K6 N$ ^: P7 o! iQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
1 _( Y+ t: q% o1 T& R[
    <$.1^2 - $.1 + 1, 1>
]

" F' D: h+ Z  M2 ]; aR.<x> = QQ[]
F5 = factor(x^10 - 1)
6 ?, V. Y1 o: W$ x, yF5
- j1 a8 ?5 s5 b(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)
$ S  z% W$ X! A( g
Q<x> := QuadraticField(10);Q;
% V2 V, x, |. y1 V- S4 F1 c8 n$ OC:=CyclotomicField(10);C;
2 }0 |! ^4 g5 B9 {6 LFF:=CyclotomicPolynomial(10);FF;

: B% x4 i- q& \9 m: x3 x, DF := QuadraticField(10);
: e  F% x) ]. pF;
D:=Factorization(FF) ;D;
. S3 i/ ^4 K" m5 ]( V8 AQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
2 _# D7 t7 F, F) A* J. FCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
/ ?8 y% D, t! n+ `  j& u7 TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
, X+ ~" j+ X) |- V& B: r, j/ w    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]