实二次域(5/50)例2

lilianjie

5 d/ k' d" I0 O& j  v( [1 |4 q$ b* q
Q5:=QuadraticField(5) ;
! E  E* P1 a- o& E9 [& BQ5;
5 b# `! b9 M9 l6 YQ<w> :=PolynomialRing(Q5);Q;

EquationOrder(Q5);
8 h1 t. a# Y5 m/ K2 u1 L! s$ \M:=MaximalOrder(Q5) ;
& H. [6 G" w( ?M;
) w$ U- `$ b/ _0 h3 J: ^0 N" WNumberField(M);
1 V1 _3 V) A, ?  l6 [6 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);
Factorization(w^2-3);
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
0 Q2 w6 h5 W7 @* _' l9 U" L% P3 cConductor(Q5) ;
0 ?' P% h+ l4 l! VName(Q5, 1);
Name(M, 1);
' Z1 f9 m3 ^. p2 Q3 ~7 K- cConductor(M);
8 N/ f9 B+ j9 |2 E, S% NClassGroup(Q5) ;
0 n2 i0 q% e4 R' h* G9 NClassGroup(M);
6 i  k* ^3 {( B) s9 [9 _& @ClassNumber(Q5) ;
4 X6 j9 R, P, ?6 ?9 F# b* S6 @ClassNumber(M) ;

PicardGroup(M) ;
3 e/ ?" c0 m/ U7 m2 x" T+ l3 N- CPicardNumber(M) ;

" s! L$ q* n3 x% z9 Z4 c
1 Q* w' ?4 w9 p# h% BQuadraticClassGroupTwoPart(Q5);
7 f1 I& k( }# ]) R$ P  cQuadraticClassGroupTwoPart(M);
2 u* d9 ?' t: L$ X. [" T

5 c5 ]* z0 K! o1 b  DNormEquation(Q5, 5) ;
NormEquation(M, 5) ;


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
% N3 F( q# w# f4 QMaximal 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 $.1^2 - 5 over the Rational Field
true Maximal Order of Q5
: r. K0 v/ D' f& i! C: q6 qtrue Order of conductor 16 in Q5
! P0 H& g% P5 h8 K2 s5 dtrue Order of conductor 625 in Q5
4 K: k, s& C* I! Q( Xtrue Order of conductor 391736900121876544 in Q5
1 l( C& C2 n1 A3 N! B$ D5 C1 z% W[
8 V1 B  v' A7 F! I# I    <w^2 - 3, 1>
]
/ |1 s+ C3 m' I. C% O8 U5
7 s/ E1 y* o( D0 h/ t1/2*(-Q5.1 + 1)
5 f3 d) l1 j  f" t1 e- u4 k* e-$.2 + 1
9 F5 a3 d' N5 f% c; R" z5
Q5.1
$.2
8 _3 p7 `- C1 i1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
6 p  S/ Q2 N# r& \. {' M1
Abelian Group of order 1
/ p, z& \* r$ bMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
3 n2 J  X: A. S* q+ ~. x$ K1 D, i  ^inverse]
/ T* n* j4 M" r7 G# d1
0 G- a% N, R. m2 KAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
3 N  @. X! P- R2 N5 given by a rule [no inverse]
Abelian Group of order 1
4 F! A, I; b  A) J! q. F. pMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% B# E$ J: ~2 q# s9 g5 given by a rule [no inverse]
true [ 1/2*(Q5.1 + 5) ]
true [ -2*$.2 + 1 ]


% U% Y% j. M' @$ A( t

# K8 M9 {4 x9 o# t/ k
& ]% t' L6 ?0 P: e& I
  t+ h, B3 O+ f1 H. a% ^

$ `6 z5 F9 m5 M% C0 O


$ @% A: R8 T% V- N+ S  Y/ N) p! N==============
' {) H7 k9 U5 G- N" Q  }9 l
Q5:=QuadraticField(50) ;
Q5;
2 d# D4 m. G1 K( n9 V& W5 S
Q<w> :=PolynomialRing(Q5);Q;
+ Z# X7 e9 @1 N9 F6 K/ W0 x9 B( IEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
( u$ J5 O+ c1 `+ j+ [8 YNumberField(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);
5 M) r; d( R0 Z# o! H) W0 i' v. xIsQuadratic(S1);
- i+ M' h# @5 n% ~& OIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2-50);  
! s7 a! O8 Q5 R' c# ~Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
! l, `2 B$ \  n/ E' ]
Name(M, 50);
Conductor(M);
: f. c) ?$ T/ L" C7 O1 @5 uClassGroup(Q5) ;
3 x' F2 k1 D2 m* u6 W0 W5 v4 C- SClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
( P: ?) {2 p/ t- U: Y, j; FPicardGroup(M) ;
PicardNumber(M) ;
" s; R8 f. j; b4 i* A
( d7 B) D4 K+ D# r: `1 i" _% dQuadraticClassGroupTwoPart(Q5);
% P* _6 A7 ~4 k+ j- ^) h, t% tQuadraticClassGroupTwoPart(M);
NormEquation(Q5, 50) ;
; i- d) {6 R( c( y2 |* ]NormEquation(M, 50) ;

% ]3 E0 c2 I) dQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% m, q+ x# @% D; dUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
' [) @4 d" Q# X  ?+ x8 [6 S8 KMaximal Equation Order of Q5
' H' {. M, l  Y4 BQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
  Z8 H; a; W% @' B6 A$ F7 BOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
true Maximal Equation Order of Q5
/ `: G8 T: U& ], `  E& Ytrue Order of conductor 1 in Q5
$ a7 j# ]$ _4 d2 Itrue Order of conductor 1 in Q5
3 X0 V2 Q* i# Y9 ztrue Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
' ]! C* n7 X0 t- M- m) t    <w + 5*Q5.1, 1>
]
7 L" u/ W7 H" ]  ?4 [" A8
0 }/ X! d5 _' P; K; `Q5.1 + 1
# C2 \$ z5 c  J1 A: d$.2 + 1
1 I$ S/ \& R* b) \: ^+ X8

3 @2 j1 e1 J+ T( J9 M' h>> Name(M, 50);
  x- a. J  a. Z" Q+ a" |: Z7 U3 ?       ^
7 I% W1 }+ S5 x0 s. k& e, C$ IRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
/ m0 U: N1 s4 ~4 I
1
  S6 \1 r+ a8 m  OAbelian Group of order 1
4 k& Y; M* R. p. iMapping from: Abelian Group of order 1 to Set of ideals of M
7 G" L) K- U8 I- [Abelian Group of order 1
  O, p" b0 [1 I- RMapping from: Abelian Group of order 1 to Set of ideals of M
1
/ a+ ^6 b8 f/ W5 t" D$ ]! X1
Abelian Group of order 1
8 }* E$ n6 S6 l) u4 F! N: l% VMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
! c+ \. X/ e" Q) R, p( Q- m0 b: Yinverse]
1
- C7 h, s( J% K8 d- z0 \3 ~* z$ LAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
' Z4 C3 N1 V2 s4 b, \9 c5 |8 given by a rule [no inverse]
' A" ^0 O! C, K' M4 _5 Z( zAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
7 S/ [( Q5 E5 Y8 given by a rule [no inverse]
true [ 5*Q5.1 + 10 ]
true [ -5*$.2 ]

lilianjie

lilianjie 发表于 2012-1-9 20:44

( Q9 a7 L. q" x6 J6 z分圆多项式总是原多项式因子：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

: G7 k: V! |! J7 g分圆域：
' Y4 `2 A+ t3 u- k4 e( |分圆域：123

- G! i- b* [1 ~) n5 T; J9 X' iR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
9 x3 |/ ]! J: N: x5 Z! N
(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;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
$ Q- ?0 i# T% m  b3 K0 @  @D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
. _6 n" e) }) z- _' l" H; Y, i! ^# ZCyclotomic Field of order 8 and degree 4
$.1^4 + 1
2 V* e5 t3 l5 b/ SQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
6 @1 W1 A: P" V0 v    <$.1^4 + 1, 1>
- ]4 y+ V2 n0 |2 U]
: P! Y0 i# J) U- B- {
7 Y4 z' Z1 T( s8 m- h. Y( c  h& tR.<x> = QQ[]
- l) ]3 ~0 z- V! k- S& {F6 = factor(x^6 - 1)
F6
8 W) Z3 E- b$ [6 ?
5 e1 C: B% H, u; X, I' e6 r! 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)
" m; V5 L# X5 X3 k
' K$ b" n% Y6 }$ z  z. a4 Y1 Q7 S6 \& |Q<x> := QuadraticField(6);Q;
# Z+ x3 p) W! Y) q5 `- UC:=CyclotomicField(6);C;
4 [( G4 y; i5 x! MFF:=CyclotomicPolynomial(6);FF;

- }2 k; B  ~: J8 b+ K* ^" @# f' }F := QuadraticField(6);
; F( ^# q0 K% j5 K, X4 XF;
# Q0 B# a, \5 CD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
! B2 S, J' V3 G, s, J$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
7 w! ?  n- x* h; D7 U[
" Y$ O: r' z! A1 G    <$.1^2 - $.1 + 1, 1>
/ f) q- n4 W# T& A1 E]

R.<x> = QQ[]
F5 = factor(x^10 - 1)
# t# l/ a- p, h/ y6 c) z  {- |+ yF5
4 g3 ~3 @) c! Y( P) n. j(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)
5 a$ Q2 y9 B1 u0 o
Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
! b  _! q4 c3 N) A- T3 o6 z, @3 [FF:=CyclotomicPolynomial(10);FF;
& Z+ x% t1 G* }, E$ T1 ~8 @; R1 w
1 V+ m& t2 Y, \' L) oF := QuadraticField(10);
# c5 j. C7 G3 A' FF;
, V8 p  m' J6 U( d+ SD:=Factorization(FF) ;D;
# }4 [) O/ K) PQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
! v% ]: _9 i- i8 s5 f) v* zCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
1 W8 q5 o/ J( H# q    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
/ r0 ?8 j; P; D( {. j]

孤寂冷逍遥

lilianjie1

) j2 y% ^4 h; o* Y& I3 o4 ^* u
8 g% F1 h- Z4 h. n( ]- V4 l判别式计算Discriminant
0 x) Q# Y9 O# Y+ s. ~
5MOD 4=1

( K* j( `  V) b+ J8 Y6 J- Z  |9 n2 d5 p（1+1）/2=1          （1-1）/2=0
' |( R4 P7 ?# X5 h5 s! ?
" V& W! H' T+ f; t+ B+ M% T; iD=5
: K1 ?- ?" R% R
" N4 V0 i" Z2 f2 Z4 s
2 H; M- ?+ D5 J- R# o- Q50MOD 4=2
D=2*4=8

lilianjie1

lilianjie 发表于 2012-1-4 18:31
4 B! u' ?* H4 U1 R基本单位fundamentalunit ：
5 mod4 =1                              50 mod 4=2

基本单位fundamentalunit

lilianjie

, b- _2 F" ]! }' z# u# i! ]
基本单位计算fundamentalunit ：
/ l, @; y8 n0 \4 [8 Y0 G2 h4 k5 mod4 =1                                              50 mod 4=2
( k, @3 G1 b0 i' U# d; O
# Q4 L) O: H% \: [ x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.


7 r+ n$ O6 N% M- M/ y/ @最小整解（±2,±1）                              最小整解（±7,±1）
5 \* c% u; B- B7 F9 U' y                                                             ±7 MOD2=1
" ]5 p6 I2 J/ m% V6 N
两个基本单位：

lilianjie

二次域上的分歧理论