实二次域(5/50)例2

lilianjie

3 E; w' B9 S1 t" \9 D+ eQ5:=QuadraticField(5) ;
Q5;
$ w1 N: `5 ~9 l9 E* q- m) R2 lQ<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
# H, z. A: F3 ^+ G/ i/ u- yNumberField(M);
4 L1 S( I; W/ x* {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;
, S  V& i5 K( {IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
Factorization(w^2-3);
9 T. X8 [6 p4 h& R4 X2 v! S8 vDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
7 [: w  [1 e# o5 _0 l" DFundamentalUnit(M);
! ~$ _+ a0 Y! s) }. BConductor(Q5) ;
6 V& `- ^, a/ n! LName(Q5, 1);
/ a2 c$ s6 B* q( xName(M, 1);
Conductor(M);
ClassGroup(Q5) ;
: Y/ t& B; Y5 t! ^( {ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;

- ^$ q$ }; u) ~$ MPicardGroup(M) ;
( T% p& B5 g  v0 ]( U7 r1 [PicardNumber(M) ;


QuadraticClassGroupTwoPart(Q5);
; \+ p1 Q4 k& q* qQuadraticClassGroupTwoPart(M);

8 F  \- \% z# ^0 e: o1 D
$ d9 m( d4 b. `2 }* `NormEquation(Q5, 5) ;
4 ~& i9 A; q: FNormEquation(M, 5) ;
+ U( i! Q5 ?" l; }/ U6 o
  x5 _4 c2 a: g! s
: F' @+ x, e- |% j1 r+ DQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
# h4 x$ b; M+ K* d3 oMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
9 k% m; `2 s) ^/ U6 jOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
: v0 k3 ^- e& q% e, v. Etrue Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
1 @) Z  H2 s2 u7 B& S/ ~    <w^2 - 3, 1>
. O- L& j6 \, n]
5
% f: u3 F& K9 G; G7 _- A' Y: z1/2*(-Q5.1 + 1)
-$.2 + 1
5
Q5.1
# N0 B, @# `, P% J/ U- C2 }$.2
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
3 y, c& X& P/ _5 i1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
0 v. y# u/ f- ^4 O/ {) ^6 Vinverse]
1
7 C( k) O8 t; I- ?0 S, Y4 dAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
) b. c' p; c* ?% u! Y5 given by a rule [no inverse]
: n: f' V5 @8 B8 DAbelian Group of order 1
2 F! p/ a2 l! G8 g! F* BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 given by a rule [no inverse]
. ?( A$ Y9 W6 ?' ?: D% z1 y* d) y; @% ztrue [ 1/2*(Q5.1 + 5) ]
4 f0 S7 ]9 q8 _2 M: N$ N, C% ]# l3 D. rtrue [ -2*$.2 + 1 ]




( ?7 A& c7 o0 u3 K+ k0 |


4 M/ F+ q& V* J. z9 E1 o
1 G. y* w& o* Y! i& k. l
; Y- L0 x. }# D% u
# o- D. t# _. [  ]! A1 v
==============
9 X/ R& }& {6 t
Q5:=QuadraticField(50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
  I! B: H' _0 {, J2 _: `EquationOrder(Q5);
7 a" U7 l# @1 o* y, ~M:=MaximalOrder(Q5) ;
/ F/ |# C$ j4 I. AM;
NumberField(M);
" f: b. M0 K1 X- G3 }5 _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);
* T1 E+ s0 z" W- BIsQuadratic(S625888888);
Factorization(w^2-50);  
( X" x! Z/ x8 n. G( UDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
' y, K) ^2 \! Y% S% U  q: I5 RFundamentalUnit(M);
Conductor(Q5) ;
' i+ k4 _7 [" G$ r' ~# H" S7 _
Name(M, 50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
' I1 T- y! U2 Y* _; FClassNumber(M) ;
9 R3 r- c+ Q" R) h! oPicardGroup(M) ;
PicardNumber(M) ;

% I  ]! o3 Q, k2 ^# s8 ZQuadraticClassGroupTwoPart(Q5);
1 O4 u3 g& p2 ^5 l; UQuadraticClassGroupTwoPart(M);
- D) Z  I$ C" G, S5 @NormEquation(Q5, 50) ;
; Y& t. b0 h( E, c3 [NormEquation(M, 50) ;

2 G, L& f  p' [# d: `2 a8 ^! L2 cQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
. @6 g- o) b* s" zUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
/ |, @4 I5 q! z; JMaximal Equation Order of Q5
8 |4 v1 _2 ~( C$ `0 ~! MQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! g  u) u2 v& H/ x5 `$ u& pOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
3 E+ ~5 ]3 E- I; P, Z5 `: Ltrue Maximal Equation Order of Q5
8 n: k+ T7 u5 U1 R* B5 utrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
( w. @: m  N4 F8 N/ k    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
8
# d9 _9 M" R. }( lQ5.1 + 1
$.2 + 1
/ o1 R* g( N; J/ \3 U) r8

>> Name(M, 50);
       ^
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
/ q- g/ y) X3 L3 C* |# V
( m/ M4 F* y# N; @1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
, M& A6 |: Z. w+ U$ nMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
8 L) ?) `3 g) s1 winverse]
; b( ]$ ~; e* O* Q6 N2 c1
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]
Abelian Group of order 1
0 q" c1 o, U. h8 C; ^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
* E" j$ P4 B# k& f2 p/ Q: F
x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.

% ]* j* |* Y$ h* ^* ]! 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 ：
  [  k! J" X- J# V6 a  X5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie1

判别式计算Discriminant
- z- H3 H& z/ o; Z7 O
5MOD 4=1
6 X) O" r! l/ J! K& H  U0 q
（1+1）/2=1          （1-1）/2=0

6 ~9 H$ \4 V- b1 @D=5


50MOD 4=2
: Z) U- J8 T* ^! L- XD=2*4=8

孤寂冷逍遥

lilianjie

lilianjie 发表于 2012-1-9 20:44

5 ?+ C; r! a9 E分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
: y9 \6 k; c! {4 lCyclotomicPolynomial(5);

4 s# @! m$ W8 L" q2 U8 G) ~! q8 F1 J分圆域：
分圆域：123
- x! ]5 t4 f. ]- q# @4 y9 D  ~+ T1 l
( P6 P) k6 e( A1 E3 J; u2 J- HR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
3 {/ l3 l2 _5 E' W8 t# w2 v
(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;
. \! g9 K' G" n9 BC:=CyclotomicField(8);C;
/ k+ m& u; ?& nFF:=CyclotomicPolynomial(8);FF;

9 N; H3 y' u7 f5 MF := QuadraticField(8);
F;
1 @: `; g1 a$ }) L! }7 GD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
$ L; W$ q: \# ?2 b6 q& B: _  N# w4 T$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
4 ^. N3 Y- L) T. n    <$.1^4 + 1, 1>
- P' z. n* L  N6 V5 []
: e& Z" W2 V3 K% d$ |
6 l9 E. j- g7 V" P8 X/ m& m2 rR.<x> = QQ[]
F6 = factor(x^6 - 1)
* z" v$ `3 S1 [& N0 ?) H) TF6
( P2 d) G5 i" w- o6 n, t- P
3 E: C. F1 `: ~6 |9 l% `2 k; o(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
- E& p" i' `/ ]) V7 [
' d4 W4 s; R6 s/ Z8 k' U2 ^Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
5 R) G+ L4 W  ]FF:=CyclotomicPolynomial(6);FF;
8 U' H& o8 @& U( d( F
7 C. V7 i9 }- B+ AF := QuadraticField(6);
2 m& B- Q! z- g4 CF;
D:=Factorization(FF) ;D;
) D3 J% m3 \* E! C9 {5 Z/ nQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
( V% x8 D7 w/ B! [# Z& r$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
$ Z( s. D" U" J) {: E2 f6 H2 n[
' ]5 Y0 @1 M+ h; O/ A    <$.1^2 - $.1 + 1, 1>
]
) t! j6 A3 J' V
R.<x> = QQ[]
F5 = factor(x^10 - 1)
$ X" t" t; a/ e$ G/ M3 Y4 bF5
: h/ ~9 K. i4 u& f" u(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)
% h/ S: K4 [1 x
, G; T8 I4 s* u7 V. n+ R& xQ<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
# S6 P' v% B, o- @FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
$ j. ^/ D  P' i5 BF;
D:=Factorization(FF) ;D;
3 G/ `' F* }+ h3 [Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
% e7 g5 m$ G2 _" K: l$ ?; @$ O# C$.1^4 - $.1^3 + $.1^2 - $.1 + 1
/ M3 q" t$ Q8 }7 k* e. n# CQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]