实二次域(5/50)例2

lilianjie

Q5:=QuadraticField(5) ;
Q5;
Q<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
/ B2 Z. \7 T/ ?: L; l  @3 c& u: {M:=MaximalOrder(Q5) ;
M;
) w/ P9 ?8 l2 ~5 C9 g5 QNumberField(M);
9 F2 R  c: d& @( H2 IS1:=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);
5 e" T' a$ v; J9 p8 [+ f$ o8 M& XDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
2 N3 o' t' t6 h! w& t4 o' r2 qFundamentalUnit(M);
3 g6 P/ X- J9 L1 k% g: o) M7 lConductor(Q5) ;
/ o9 b  Q$ D$ m, sName(Q5, 1);
Name(M, 1);
Conductor(M);
7 J' e4 M, Q: Z2 rClassGroup(Q5) ;
% |' |4 ?' t1 `) q  z) S# j  ]ClassGroup(M);
ClassNumber(Q5) ;
7 u0 w5 |- R7 j" u2 ?* R6 y; ?1 _ClassNumber(M) ;

PicardGroup(M) ;
PicardNumber(M) ;
! J6 M) h5 F, l

( C4 C: ^/ `5 N4 ]& |3 GQuadraticClassGroupTwoPart(Q5);
  \% K9 x( b# w$ H# qQuadraticClassGroupTwoPart(M);
' l! X9 k( L! }- k
5 I: h9 _# I5 V
NormEquation(Q5, 5) ;
NormEquation(M, 5) ;


Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
  S/ p- m& t6 w, Z3 [* A2 eEquation Order of conductor 2 in Q5
Maximal Order of Q5
% ^0 m0 r8 u- {9 C8 H3 d7 [Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Order of conductor 625888888 in Q5
9 W8 q7 Q) F, T& g; Ftrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
& d" J$ Z  w9 ~: Q' B/ I. ktrue Order of conductor 16 in Q5
' o0 Y) h6 s/ btrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
: R; z' T* L) W6 ][
% Q% |* \5 q; I) ]. g! C    <w^2 - 3, 1>
]
1 I' i8 Y/ u/ C( `- p5
! E. {: {) l7 l. s1/2*(-Q5.1 + 1)
-$.2 + 1
' l; U* i. V" d( Q. Y5
7 f5 V* j& T. C+ V. yQ5.1
* }" J- r+ x; z5 C- ?1 R$.2
1
Abelian Group of order 1
' z: R8 h) U5 p9 J* uMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
4 K+ u8 }* c8 `( E: P* r% RMapping from: Abelian Group of order 1 to Set of ideals of M
5 @) V; ~' k. F1
0 _: k4 _1 h  K) E* c% K, J' D* N1
Abelian Group of order 1
# t0 Y( c' [( h& b: Z) uMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
  P! j' |& Q& \$ t  p, R& k1
( b  F- K; X2 ]1 V% J6 Y0 ]& cAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
# Y0 m1 f5 B" k; B* ?" ?5 given by a rule [no inverse]
Abelian Group of order 1
3 Z7 o, A" C% V( u" e3 UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 T& T/ v: W* `/ j, X- T1 {5 given by a rule [no inverse]
: Z+ T- I% \; s$ j+ }true [ 1/2*(Q5.1 + 5) ]
8 H5 L$ F" ~+ Rtrue [ -2*$.2 + 1 ]


+ G$ A# H5 S- I0 I: N6 F

# W/ V- y" S/ G, ~! G7 W
9 E2 X2 s" g$ ?



+ t$ X3 C, J6 ?" U% R
# N- N3 l, ], i0 L) P
, U- A4 O5 c4 }4 w; z; p" g==============

Q5:=QuadraticField(50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
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);
: }* z( k1 M( _+ ~' ?' y1 V1 uIsQuadratic(S1);
4 U9 g5 f5 K5 L  H. g8 R! c+ f$ FIsQuadratic(S4);
8 C/ m9 [3 g( {$ b' \( `3 L7 TIsQuadratic(S25);
2 B: H- C* j; [- l6 `$ r' ]* L6 F2 DIsQuadratic(S625888888);
; w- x. a. J8 M" ^1 [  nFactorization(w^2-50);  
( [% s  D9 z+ Q) t. FDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
) L. S( V9 b- P5 y, |FundamentalUnit(M);
Conductor(Q5) ;
  b  @5 A- `# M1 l$ f
Name(M, 50);
Conductor(M);
ClassGroup(Q5) ;
$ _/ M+ I! `4 HClassGroup(M);
( Y) ?# ?! h% NClassNumber(Q5) ;
' @4 S8 r4 O: b- _+ ^1 i5 j% h. RClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
- D. W+ t' v2 K- k, z* C3 kQuadraticClassGroupTwoPart(M);
2 {# M8 d4 E- [4 dNormEquation(Q5, 50) ;
NormEquation(M, 50) ;
) V% v* Y! N3 q' a; G2 B3 [
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
: f3 \5 \4 g' q/ DEquation Order of conductor 1 in Q5
: Y4 l/ t; y6 o/ X* U6 c; G  r7 P, NMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Order of conductor 625888888 in Q5
3 g0 I. x. n. k6 S) i) htrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
5 o. j( e: A7 B* Z& qtrue Maximal Equation Order of Q5
' y( Q! h  d; X  etrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
2 F  h8 a9 M' N" |) S9 o1 g  Btrue Order of conductor 1 in Q5
# G& B8 U6 Z: [/ y8 H: q( o$ h[
    <w - 5*Q5.1, 1>,
# t. q1 C+ Q: A5 \    <w + 5*Q5.1, 1>
6 Z( l; |! M) l8 B]
& O2 S+ n: T" R+ j& Q9 n8
: \( k0 `# ^9 z/ a. p3 M$ kQ5.1 + 1
. X9 Q+ R. A3 ^; J7 u1 a, v( Z$.2 + 1
8

>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
; s& }+ a( K( T! V7 z4 R9 d
% x/ D; r4 h$ r# y: b1
  _/ W2 y0 e4 T6 Z  H, D' d* O& CAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
' J+ ^* X' o8 J9 R$ P& ]6 KAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
) K+ F7 v8 j+ u- Q6 R2 ]1
Abelian Group of order 1
5 F6 w* s& W3 {2 E. Z+ h7 YMapping 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
6 L7 A( x& L* q+ t& Q! t8 given by a rule [no inverse]
1 }8 ]; g% W# [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 ]
true [ -5*$.2 ]

lilianjie

二次域上的分歧理论

lilianjie

基本单位计算fundamentalunit ：
5 mod4 =1                                              50 mod 4=2

4 s& J1 t) o  D3 ?$ R# u/ z( z7 b x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
1 d" g% j' a) z3 t' m1 F2 g' d x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


3 \) Q( B4 t# ^! W3 k( z最小整解（±2,±1）                              最小整解（±7,±1）
, H+ s8 E+ @- ~5 l                                                             ±7 MOD2=1
, s' F7 }- `# S  ?8 P. Z
0 \, ~5 S) Q- P+ q! ^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

* [  d- n( M+ K# e; l2 ^6 h基本单位fundamentalunit

lilianjie1

0 L5 E9 Q2 Z' ]
判别式计算Discriminant
! M" r$ i8 {; C! E
, G. l2 ^! j% G5 w% w5MOD 4=1
- L/ J7 C* }" w/ K6 X
（1+1）/2=1          （1-1）/2=0
5 _6 F' A% W2 X9 P
7 T; t  N- r. D. m( t: }; h; B$ a. BD=5
  c! f" }4 v. G3 N6 |+ _7 M
; O3 ^5 h& c# h/ l% Q
50MOD 4=2
5 [7 u5 E% E' {, I1 `! L5 F8 P5 lD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44
4 b5 E. U7 u2 \8 |6 x: O* {
7 b5 t2 r: _/ n$ ~! c$ v& ?! O分圆多项式总是原多项式因子：
. m5 f+ V( l/ M. K2 X3 CC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

( J5 W2 [% @% I( [7 b3 v$ V分圆域：
6 q, \  u4 Y6 U! G) u* s0 j/ l分圆域：123

R.<x> = Q[]
( U* G5 c+ [; k' M+ `. ^1 V4 kF8 = factor(x^8 - 1)
# c- s) ~0 \1 F- C7 @F8

- I* g7 D' C# N: i3 _* r! Q(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
9 P, x; C5 I  z- P* m+ _+ h: H
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
* _! r3 c# [/ U# j& \  \$ MFF:=CyclotomicPolynomial(8);FF;
/ U/ }; d) ~8 A: e( f* R* c( m. v6 N
$ m' R3 ^5 F3 dF := QuadraticField(8);
( v! [6 @8 C' w% h6 k3 YF;
D:=Factorization(FF) ;D;
' ?' X( }3 p, MQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
7 z* {" R: n" \, r, W, T0 m" V& {Cyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
, S* `6 Z$ `) H; i# ?( N, G6 o' `    <$.1^4 + 1, 1>
]
" P' J) P0 Z; ?* v. r+ [# Q
4 T) T/ Y( \* }. X( w6 s+ O, E9 XR.<x> = QQ[]
8 U; r- L( H4 @( |F6 = factor(x^6 - 1)
F6

3 k) t: I4 e2 j) Z(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
3 N; \3 N6 }% m/ A9 [$ o
, r) [. P5 m0 M; x) a5 TQ<x> := QuadraticField(6);Q;
% w! T" ^/ S# O7 |7 J. tC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
* c/ ]' [5 Z; A* \3 S" _
3 W. ]. c. n$ U" ^F := QuadraticField(6);
F;
: T1 N- N$ e- nD:=Factorization(FF) ;D;
6 u- C- ~+ C5 v; d8 y  w( N. _3 ]Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
% d  F6 o$ T. t( N3 KQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

, C8 h% ~6 |0 W: p" X6 nR.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
) {. Y) f6 G) I0 ~(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
; g2 V" q  G5 d; ?/ A. Y1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

5 R, @  [0 J. E# H+ ?# ^2 i- v& DQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
' ~0 i6 c# X0 u  s, BFF:=CyclotomicPolynomial(10);FF;

# x7 `1 L6 w7 [$ EF := QuadraticField(10);
) ^( I. X6 P) I7 C8 T  C: yF;
9 ~* X5 V7 c5 B9 u4 f" jD:=Factorization(FF) ;D;
: I4 V9 J4 |: ?3 wQuadratic 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 $.1^2 - 10 over the Rational Field
[
+ [6 z8 x0 ~, i+ Y& T    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]