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标题: 实二次域(5/50)例2 [打印本页]
作者: lilianjie    时间: 2012-1-4 14:05
标题: 实二次域(5/50)例2
本帖最后由 lilianjie 于 2012-1-4 17:59 编辑
1 q, z+ \" _4 Q; ]' E; \8 i) k" v+ k
Q5:=QuadraticField(5) ;
( d3 |( d, X# F4 G4 C! C' Z, c7 QQ5;& U! [  f4 i% `* `7 C, c
Q<w> :=PolynomialRing(Q5);Q;+ {2 \% q) o* p4 p. D

" ~6 A- z9 a$ a9 a5 r1 j( v, XEquationOrder(Q5);
( J3 C* v7 J; cM:=MaximalOrder(Q5) ;- N" P) W& z, m7 {
M;
4 p) N( w* G. M, ~$ E- L) rNumberField(M);
. I+ r0 H6 s5 {2 w4 }1 eS1:=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;
' h) Q" @0 K2 @" i2 _% NIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);4 u- a6 u- w3 R& n
Factorization(w^2-3);; C% p/ y2 I4 T2 @
Discriminant(Q5) ;
8 c8 p4 x1 U$ o/ r8 t( U/ L, PFundamentalUnit(Q5) ;! G3 G$ q1 \: Y) r  a5 b
FundamentalUnit(M);
' h3 O) B/ S0 _  I8 S. n5 hConductor(Q5) ;
$ d0 J/ a: R  Z0 u0 d* m' ~Name(Q5, 1);1 `8 P& f8 j& K2 P- I
Name(M, 1);9 b2 \( p2 H2 a% v4 b
Conductor(M);# |0 V8 [7 p2 e
ClassGroup(Q5) ;2 `# e2 ^+ ^& |2 `' W
ClassGroup(M);
7 ~) S5 G2 q% }8 ~5 u/ NClassNumber(Q5) ;! q8 B( R( G7 K* o5 O% R
ClassNumber(M) ;/ k4 i+ a6 J) p5 d2 a: z

3 n: a$ A1 `  K/ DPicardGroup(M) ;3 s: d. J' a+ s6 Z* U
PicardNumber(M) ;
7 `1 A' f9 V" P" g0 g
. @" b) g, ^: e& y! {8 D. _) Z( Q
1 ^' ]: ~; n/ Z' ]; xQuadraticClassGroupTwoPart(Q5);
& v1 G) d( A: p- vQuadraticClassGroupTwoPart(M);
- O0 G; d; M0 O% s- \$ A+ T: {% |& x: Q/ R

, d  Y( f( V& s+ N, n9 i+ tNormEquation(Q5, 5) ;
5 `% d$ `  R/ hNormEquation(M, 5) ;( }/ _1 Z6 t4 E# B6 p

; l9 ~9 v3 p4 [- k7 ?+ G, Z% K' C/ ^& {1 j2 I! n
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field. e& G7 u" x; r
Univariate Polynomial Ring in w over Q5
+ [; M: U. D6 ^$ H8 t7 l0 ~Equation Order of conductor 2 in Q5* |0 O3 n! S# w7 q, h
Maximal Order of Q5& U" G1 o8 L) K# Z) w
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field: P  Y) b/ M. @5 d3 o, U
Order of conductor 625888888 in Q53 g  [3 \5 p6 }/ o8 k4 e
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
/ `$ O: E4 K1 E! L: N- @& \- M( k6 ctrue Maximal Order of Q5
- S9 G8 B' h; R. C) w8 ]true Order of conductor 16 in Q51 [: Q; p+ K7 V- S; v
true Order of conductor 625 in Q5
0 N4 {& C8 f' w3 t- U$ strue Order of conductor 391736900121876544 in Q5! C! b, ?" Q* W
[; F% m9 e( z8 ~. A. x
    <w^2 - 3, 1>
2 @2 h# J& M* ^% c9 M" P) W7 y]
3 M; ~. @1 r. S2 x! I- F/ v0 {6 o5+ P% o, L/ U5 K
1/2*(-Q5.1 + 1)3 ?3 p2 \  E5 g. J
-$.2 + 1' h8 l  H0 V# C8 c+ {4 G4 }
5
) C' H( T+ _0 NQ5.1
+ k3 S0 a3 X: J  a8 R! y/ N3 K6 \" W$.2
6 `. c5 E, g" C6 h9 w8 o3 a1
% V9 H+ Y& s2 N4 ^2 IAbelian Group of order 1
- }1 o- E# w/ g4 s1 T2 g+ rMapping from: Abelian Group of order 1 to Set of ideals of M
$ Z- O$ d4 T! h3 p$ T) j  fAbelian Group of order 1' C" Q7 k& J  ?' a: [* f
Mapping from: Abelian Group of order 1 to Set of ideals of M
$ q1 Z  C8 c! y- i6 R% A1
; C7 n0 R$ K0 }4 d0 x9 f2 n1# A7 L, k0 N5 Z1 l3 ?
Abelian Group of order 1
; n4 I+ d  @& ~% L* F! _& gMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no/ q5 F0 a7 |4 R
inverse]" Q8 `: v( v" S+ x7 ?! ^1 M2 ~
1$ K8 y3 Z5 F) B( i* c+ v5 d
Abelian Group of order 1
8 `5 g: x# B5 m' P4 XMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant9 Z2 }8 }% |5 z, a0 a
5 given by a rule [no inverse]
' [1 g0 S5 g9 a6 }( o+ VAbelian Group of order 1& G8 b7 s6 }* j7 O
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant- D# U* Z0 k7 {& p  `
5 given by a rule [no inverse]
; b; p1 p- |' l8 N2 r; Ntrue [ 1/2*(Q5.1 + 5) ]4 E$ D& z8 v8 D0 s1 M
true [ -2*$.2 + 1 ]! f5 Z# h# S' Z1 T- p9 `! D5 j) `- q

  f6 L! F# F4 G1 n6 `
: g! v! I, }- l# D8 |
" r! y$ ^/ P6 l7 ]$ Q  j0 m* R( F6 n/ W! Y8 p

$ U0 U; Y/ Z) `. H% b7 Q1 \$ I1 n4 D3 [. W
3 s) b9 j0 }+ l! g

- N% v9 R& M! r! D8 _, [; t/ m0 S
, B3 r& R4 ]% {" C/ q% J$ R5 b. |2 b* I' B# v

* p3 |: [: Y" m==============
& L- w, w: l6 M* U( z, b, r  _2 Z7 ~
Q5:=QuadraticField(50) ;# c! C2 F: h( m; N0 e. ~) h, V3 T
Q5;
6 E" O3 {: O% V  r; o( ^8 o+ D) |9 L, a
Q<w> :=PolynomialRing(Q5);Q;' u8 K' r% G8 b' L/ D
EquationOrder(Q5);
7 b, O, S0 d" p6 JM:=MaximalOrder(Q5) ;
8 G0 F6 {/ W  {) {, f- oM;5 S$ c% K# I0 h! z
NumberField(M);& \+ c3 D; C+ j/ Q
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;1 g# W8 Z0 \1 P! ]5 n5 o  V1 x
IsQuadratic(Q5);
- F0 @8 v6 u! V3 I. DIsQuadratic(S1);; q+ T& w3 G- E4 ?2 y4 R5 m, g
IsQuadratic(S4);
1 u% j' q2 K$ x. T3 Q9 \" q/ a" v1 LIsQuadratic(S25);2 u) [. D8 f+ O! F" e) ~6 _
IsQuadratic(S625888888);
1 G. u! X' _  z: M) Y. p1 fFactorization(w^2-50);  # u- U& u' G) C) w' o8 `2 G# i( N
Discriminant(Q5) ;( v; y4 D2 h5 T9 z3 @
FundamentalUnit(Q5) ;3 |+ y' Z7 `7 q/ `# c2 [
FundamentalUnit(M);
6 F0 ?! m! j' Q( G/ I0 p5 h% A( ?Conductor(Q5) ;
( `; E. b7 \$ S5 r. Y* u8 x
% o7 f! p0 n) ZName(M, 50);
6 w" ^" B1 W( Y; g/ r+ i  KConductor(M);, O+ I7 A* T; L" n# J
ClassGroup(Q5) ;
7 ]0 H" |# H- g& [; tClassGroup(M);
" W# }- x; {! A% C/ M5 WClassNumber(Q5) ;) Q& A9 j$ [9 G3 H
ClassNumber(M) ;
8 s( l! T9 J7 Y/ ePicardGroup(M) ;+ o5 }. c* ]1 i; Z
PicardNumber(M) ;
/ @" Z& _+ q3 [2 Y
, W# C/ y' @: X$ qQuadraticClassGroupTwoPart(Q5);
& b% C* K9 h% ^9 C9 lQuadraticClassGroupTwoPart(M);
8 R2 G* n: e& k5 w9 XNormEquation(Q5, 50) ;/ o: h1 ]8 h, N: F* d2 i
NormEquation(M, 50) ;
* l: b( L# n' \5 d
% F6 P" ^& O3 d6 h* [Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field# T" x! G* X  C1 w' x$ |5 n
Univariate Polynomial Ring in w over Q5* S! [, f" {+ @5 W3 H# x" u
Equation Order of conductor 1 in Q5: g$ F! \7 K" b- L/ L; @* p
Maximal Equation Order of Q5% `  E: `0 ?5 g* Z! c  V1 B3 g7 b
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field# N6 Q8 o3 n9 d* i+ m& }
Order of conductor 625888888 in Q5" h/ ~3 Q3 S$ |' |2 F
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field9 q" U/ x, G1 `* y$ m  v
true Maximal Equation Order of Q5/ V5 u2 u! c3 V6 X  P- [
true Order of conductor 1 in Q5
. ]2 ~7 U* o1 r+ t/ g. _1 ntrue Order of conductor 1 in Q57 O4 l& M3 g, {' q, |% X
true Order of conductor 1 in Q5" m8 _0 c: Z9 a! E; P+ |
[2 ]# o3 @2 d+ c. K
    <w - 5*Q5.1, 1>,7 n/ R, Z! x9 O4 n! W
    <w + 5*Q5.1, 1>
% q- D# l# d$ _9 E" I]
4 S0 M# R2 z5 d8
5 G1 B* H9 |5 d- FQ5.1 + 11 D7 T4 [' M2 H4 h+ N) N+ W
$.2 + 1/ i+ F* z+ a5 v2 g3 J" u- B" l
8
) M2 \2 O; o4 U$ i( u' G9 i4 B3 [8 W( ^* C; `) M. J
>> Name(M, 50);) o: V, T) X7 P4 P# [+ E5 e
       ^
# v  v) z8 P7 K0 B# V7 ARuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]- [$ e7 K. H+ ]: T- u

! X. A0 e5 i$ @& v1 x, v" S1& B% `* m1 W. c5 ~& K
Abelian Group of order 1
1 d0 E) L9 A  {6 V) QMapping from: Abelian Group of order 1 to Set of ideals of M& N, ^* d' E# ]7 [2 l6 m1 R6 L
Abelian Group of order 1' L9 s8 W% k6 ^6 P3 M: A
Mapping from: Abelian Group of order 1 to Set of ideals of M
0 o! k0 p( v+ F1
5 R9 w7 Q' P8 H, X& G5 y1
6 i' `7 i$ f6 z9 ?: a: g( k, SAbelian Group of order 18 M- }$ C  p, O- t2 C* N" A
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no3 c: O9 s3 w& f
inverse]
7 ]8 u5 ?# l+ _- @9 s  w; y% g, y1
5 C' \: V8 F9 u% n" U; pAbelian Group of order 1
; I/ E& v, n$ R- Q# f7 H4 W; P0 FMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant5 k: k7 j' s- c. K# i- z
8 given by a rule [no inverse]- P0 s% N' \; n
Abelian Group of order 1
( e/ o( G: P7 KMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
* Y# j) v. y& j( s7 n8 given by a rule [no inverse]2 [. e8 s& D# d- p# c. x$ h0 }
true [ 5*Q5.1 + 10 ]
" Q+ r& e) T* t  A- q2 D. Jtrue [ -5*$.2 ]
作者: lilianjie    时间: 2012-1-4 18:00
二次域上的分歧理论

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作者: lilianjie    时间: 2012-1-4 18:31
本帖最后由 lilianjie 于 2012-1-5 12:42 编辑 ; x# ]7 ~  w7 k6 D9 G

7 \% A1 W2 z. q1 o; E; w基本单位计算fundamentalunit ：
5 {) ?  F5 Q2 e! ?3 a% v5 mod4 =1                                              50 mod 4=27 @7 ~! f4 f7 D

3 `3 t# G! s4 r7 f+ A x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
: }- ?/ }: O/ c7 \ x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.  l5 B9 S, ?0 h% g, b0 Z. A) C& S

/ |% q. F% M1 w6 c( A3 h1 y  v6 g
, j1 k1 H) _1 M% v最小整解（±2,±1）                              最小整解（±7,±1）
( M% Y) h: u1 H5 I' L                                                             ±7 MOD2=1
0 [$ l4 v* F7 t9 w+ o( P4 b% }% _/ D( u8 y8 z
两个基本单位：

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作者: lilianjie1    时间: 2012-1-4 18:53
lilianjie 发表于 2012-1-4 18:31 * j& s! Y( {  j/ A
基本单位fundamentalunit ：0 D5 K6 F2 G  {, H( P$ u3 M# H$ ]
5 mod4 =1                              50 mod 4=2
5 ?# g% C- h- W. }- ]' T* n0 W, Y1 C
基本单位fundamentalunit

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作者: lilianjie1    时间: 2012-1-4 19:07
本帖最后由 lilianjie1 于 2012-1-4 19:16 编辑
( `* g: t$ d# Y/ K7 W' a" c/ ^+ Z6 H2 w& t3 Z" A
判别式计算Discriminant% @; W1 e. E2 G: }( g% v$ f5 l

# P, e! o4 h2 h) @, X/ U6 h5MOD 4=1
( x8 m& F/ _' ~
& D6 Q# O( b( s# C+ v（1+1）/2=1          （1-1）/2=0
! L$ [2 ~0 c: ~& ]0 m; e, T. @1 S
/ O7 T- `: M# J% c( r8 O  _D=5) T, `: i  [2 r" S1 g9 S

8 t+ a) o; {0 @: i+ Q: \' K- w4 _5 k6 D
50MOD 4=2* w' W' |! {' l5 t* w8 A; s9 i
D=2*4=8

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作者: 孤寂冷逍遥    时间: 2012-1-5 08:37
作者: lilianjie    时间: 2012-1-10 17:49
lilianjie 发表于 2012-1-9 20:44
' ~' o; z  {" ]# ]0 e* g4 c$ ~, |0 G! {2 C" k
分圆多项式总是原多项式因子：
8 a' g" k; @* O, Z2 s) p: f. o; iC:=CyclotomicField(5);C;
8 y7 ?( G4 }# P! B4 x& N6 aCyclotomicPolynomial(5);
: @3 a& C# ?0 D( E# J" e

2 j5 j- B4 r# ?3 Q( j' ?3 F6 K分圆域：
0 x2 ]: D8 }" U; O分圆域：123
8 h7 X9 G) m0 _6 f+ d) g
9 @' L7 @) e6 Z/ x3 w! CR.<x> = Q[], F% h& X" l# G/ W# `0 @
F8 = factor(x^8 - 1)
+ k2 X. f& h! B, v3 XF8% I$ |2 z6 Y& I( d8 l
- O1 J# C" m) @
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
- R  S' ~9 {  [; ?" M% Y
  Y. ^. u& I, C$ w" m% ?/ VQ<x> := QuadraticField(8);Q;
! M+ b8 ^. W8 u; k% HC:=CyclotomicField(8);C;+ M. F4 u( L1 J0 D! U' Q- ^, U
FF:=CyclotomicPolynomial(8);FF;9 t3 i% X3 H  k. k
/ x2 ]; [8 D( e- d8 c7 Y" J
F := QuadraticField(8);; q. t# c. [' D
F;! w, _" ^  D) {0 j
D:=Factorization(FF) ;D;. g& k) S4 ^. L) r( e
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
# r' D, ?1 N7 T! J$ g$ q8 X/ CCyclotomic Field of order 8 and degree 4
2 ]) P9 w( S, Y/ S# [+ j$.1^4 + 1
* x" u5 E" t2 D. Y6 k; H# @8 ?Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field: u- e7 C! B; o) D
[
  G+ D3 l# x8 M2 S2 V    <$.1^4 + 1, 1>- M* F. I6 E6 P; Q; p( \
]
& y/ r! U$ q0 P* e3 h+ o
& |: [6 ~% n, J2 ~% f: dR.<x> = QQ[]) \: ]0 J& z9 a3 Z% v
F6 = factor(x^6 - 1)
/ K, h& L% a- M! {" V: zF66 ~5 g& @8 b& c& z
  f: L2 U: m: L  A  ~- H
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
" |) ], ?1 m3 x( d1 {) u* W% i, l( n5 O% q
Q<x> := QuadraticField(6);Q;7 y+ U4 y3 A1 ]' x/ Y; E" t/ \, v
C:=CyclotomicField(6);C;
1 o+ x# G& k6 j2 }0 s+ I" BFF:=CyclotomicPolynomial(6);FF;
! M- a; J! j5 O6 R7 o/ n
! q6 F4 [2 S* W8 vF := QuadraticField(6);9 A* [( {  l6 @# X- g% Q1 \) E$ d
F;
& p/ }  q& u9 ^  zD:=Factorization(FF) ;D;
6 ~2 b4 \& h" V; f) r, K7 fQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
4 P1 w' b, f* Y; G( P  uCyclotomic Field of order 6 and degree 21 ^) P* D% J9 \$ k2 c
$.1^2 - $.1 + 1, _0 \5 B# J. f! v' B+ U0 ?
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field1 @# b% M( b7 x) V
[
+ C+ s4 P# R5 P: R    <$.1^2 - $.1 + 1, 1>7 ~1 n: f9 ~' f) I1 s& w
]
& Y" @, m( F# R3 q* {% v6 s+ l$ U1 M
R.<x> = QQ[]
$ V; H( @8 Q0 p7 `. T% E* q3 v% F: hF5 = factor(x^10 - 1)
% M* x# a$ d9 b6 R8 EF5: X+ T/ c) U1 z; v7 M! @
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
! M. k' `) I0 w2 J8 K; d! @5 Q1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)3 S/ W% W* D# ]

. w" E4 Q# r# X7 v1 g/ B' nQ<x> := QuadraticField(10);Q;2 ?, ^3 M& Y8 R2 w0 _
C:=CyclotomicField(10);C;
  R; u9 d  r0 ]9 \% fFF:=CyclotomicPolynomial(10);FF;
8 N  v, b, }/ ]# j- C
( r) f. i# ?. G# g9 d9 fF := QuadraticField(10);
+ ^% V7 _1 i  W" `5 tF;
( J0 \9 M2 ~, o  oD:=Factorization(FF) ;D;. n2 p7 G+ y* y% P, j7 A
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field1 y( Y% N7 x* k; a5 v  V: w3 I- Z
Cyclotomic Field of order 10 and degree 4
1 Q' l! c$ O8 k6 l9 y; \$.1^4 - $.1^3 + $.1^2 - $.1 + 1
8 }& M- p1 H# R% TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field+ U- t3 f9 k/ r5 x
[1 |# N) |  @$ Q/ ~* q# r7 [: T) _
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
0 e6 P4 g5 j$ X$ j$ }]



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