本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 1 P7 ^( e: [7 H; R3 e) i8 W" v0 u7 Y4 A
Q5:=QuadraticField(5) ;, k& I* @! Y9 z' u q
Q5; . s- D$ s4 I2 m, V Y; V" P& p7 \Q<w> :=PolynomialRing(Q5);Q; ; |& m. U9 a/ W* O! U0 u ! ?" U# H4 C: U$ ~EquationOrder(Q5);7 {! C( j7 B" A& r! Z2 _; s0 o
M:=MaximalOrder(Q5) ;# b* E& W7 y8 A1 H! D
M;) O+ D! E. p: Q8 p
NumberField(M); ( D9 ~ G* e6 b3 K; _4 k& D% D Z- q9 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; + ?' a9 U* E: G; v2 `) \IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888); / j6 c z u: W' R4 V) |1 c9 _4 hFactorization(w^2-3); 7 z3 E) b5 F* Q1 E. nDiscriminant(Q5) ; ) c- p9 q6 G+ k5 rFundamentalUnit(Q5) ;' j( A: s1 {" M( N
FundamentalUnit(M);) R4 B7 U2 b4 K& B" z7 x* e
Conductor(Q5) ;# J4 x9 R7 K& L! h8 c. E+ P
Name(Q5, 1); ( H9 `+ s0 B4 S1 WName(M, 1); 9 i' q& d e' B7 B2 jConductor(M); , i2 i$ U. O8 ~ClassGroup(Q5) ; 3 `; `+ K5 \( AClassGroup(M);& _# `- k8 ~: ~* E" T
ClassNumber(Q5) ; * L5 r# u, H0 NClassNumber(M) ;/ R V$ H/ c0 K% l
% x- r g y3 K7 }5 B+ xPicardGroup(M) ; , S: {0 t* W9 w$ @; KPicardNumber(M) ; , o k5 ?+ a5 d" u- g" O 8 O! S& B) w0 n" n F " T7 U: J6 i; E& \5 J3 O8 S2 V6 _QuadraticClassGroupTwoPart(Q5); / i7 M4 W/ ?+ v6 w9 @* \. W1 E! TQuadraticClassGroupTwoPart(M);/ {3 O, Q; ~$ E, k
5 `% I$ C. ^: x$ r4 _
+ B9 d. {! n4 x3 s/ R( UNormEquation(Q5, 5) ; 5 W4 z: C: ]; g- J, N7 |NormEquation(M, 5) ; " N% X F- }* W7 }! h6 H j6 o# A2 E* [4 x
' G" x+ [% h2 y, uQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field * m: Q n" X7 m: VUnivariate Polynomial Ring in w over Q51 }2 O4 j! V) N0 b. g' A9 S3 o
Equation Order of conductor 2 in Q5 * I% l2 b2 z" ^( Y' LMaximal Order of Q5" P% S) ]6 y0 }& \* A$ _
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field : X( F# ~+ b$ w2 r4 o* k' yOrder of conductor 625888888 in Q5: H) I+ Q6 ~% `
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field 8 T8 t( Z. Q7 k' Wtrue Maximal Order of Q5# P$ x7 k! j9 M* u6 ~; h+ H7 g
true Order of conductor 16 in Q5 : n" U4 {) o4 ytrue Order of conductor 625 in Q5 ) k" Z/ B/ ?* @- J5 Ttrue Order of conductor 391736900121876544 in Q5/ o+ {' C% m" @% I! a
[; e- P+ v' o& N @
<w^2 - 3, 1> 9 `% q! f) ^% ~) g/ f] 2 }7 Y2 Q: w8 b4 n/ ]) C5 5 r1 ~6 |7 Y4 p$ S9 i1/2*(-Q5.1 + 1) . `. \. O& ]. ~-$.2 + 1( J6 T0 E' ?8 U& V! ^! K
5 . s% R# m2 ?1 M( MQ5.1 , R. G6 a" [7 r4 p4 j$.2 7 H. h1 s& f; K3 {1' Q9 L) A4 ?8 Y' Q. A# m1 k
Abelian Group of order 1$ D, ?1 S8 i7 g3 O' {% i1 v9 d
Mapping from: Abelian Group of order 1 to Set of ideals of M, K3 x. T' \# m& A
Abelian Group of order 19 N! D; H- z6 z- S! b; C
Mapping from: Abelian Group of order 1 to Set of ideals of M 2 ^$ | i7 A/ S6 e6 ?) J" p1 0 l" ?9 ]$ X n5 k0 f! E1 G$ d9 S4 u- g' E
Abelian Group of order 1 0 Y+ e- O$ {! }( N) v: \$ C+ @Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no ; y" Z: | G6 q1 V Cinverse] 7 [9 O6 L; S6 ?# z1 5 G2 j! Y+ i2 m' j/ p, cAbelian Group of order 1+ ^9 Q8 N5 Z% w+ \% ^
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant2 @ c9 ~1 ?" N7 b3 p. w# ~& u
5 given by a rule [no inverse] , i1 n" e+ ~3 V3 U* S6 t+ Z; ^1 d) ?Abelian Group of order 1 0 k" y7 b& Y* n6 @0 ~: jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant `" W# Y$ `7 s3 D
5 given by a rule [no inverse]0 y' ?5 D" a6 B0 l+ k
true [ 1/2*(Q5.1 + 5) ] ! v( q' ~4 |) K. |0 V2 y* B3 Etrue [ -2*$.2 + 1 ]& b; j$ P/ |* v: O" P) Q, o
6 e4 [4 w2 S! ~; g2 a- [* G3 P K0 |& D; t8 D# D
! ^" V' G$ o. ~& D
* F% b7 ~3 P- p 3 l! ]3 o, E) q. K* V& O " u& i5 J# I/ x2 l- Q3 F) Q- k9 p
. i4 m4 G! e" q: z& p. q# ~& Q9 \5 a' K' Z/ n4 u& u
' H* }% N! Q0 h. p# \5 R) `# n% D* |- x
; q4 ^* s. j; R; |' H$ k" x==============) |) ^8 Z1 X, C7 k
0 d+ t! ?; H7 I, f( Q* n GQuadraticClassGroupTwoPart(Q5);4 I9 R* D- Q6 h4 o. q' F D5 h
QuadraticClassGroupTwoPart(M);6 i* T7 P0 Y* ^, D& z k- E- [
NormEquation(Q5, 50) ; 1 r5 v1 x7 ]) [- QNormEquation(M, 50) ;# I/ t/ ]6 E3 X' K- q
) `4 L2 E% t) g! k$ s7 M" y
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field$ H* c. b- M- u, S0 {; S- ?
Univariate Polynomial Ring in w over Q5 * s- y4 j# h9 O. M3 w# FEquation Order of conductor 1 in Q5/ L; {7 d' I6 M+ w
Maximal Equation Order of Q59 ~3 D! }6 d. x# o9 b
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field+ O3 h* ~/ P( b/ G
Order of conductor 625888888 in Q5 " C% d" T) ~5 v5 \true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 2 c+ I3 G" t3 [4 v* h' _ k6 @true Maximal Equation Order of Q59 q& ?1 n2 T7 ]# B" p) T
true Order of conductor 1 in Q5 4 k' j' h+ F0 }2 \% E$ {% Q, X {true Order of conductor 1 in Q5' z( z+ M! D5 J+ s+ F
true Order of conductor 1 in Q5 8 @# x- @& R" _[ % y. b8 w1 ^. C3 w9 L2 ?3 E6 X8 e <w - 5*Q5.1, 1>,. T4 m6 h0 P8 A1 h
<w + 5*Q5.1, 1>$ g9 u8 j0 v7 Z/ w
]( a/ U. U) p( J* n" z) C& [% F
87 I- u s, u$ u, z2 t
Q5.1 + 1 * k. Y! W) c; r1 V) W$.2 + 1 ; Y7 r/ Y# b6 j/ k89 x7 Z5 n( B# o# R& t
* G; ]2 U& _3 r5 t- W6 \
>> Name(M, 50);8 y! Z- ~/ v6 k! O
^ $ R8 n% ~' a( b' i4 J: cRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1] % k6 E9 ~7 b7 S% o' N ; b; i2 T$ Z' B# b1 / t0 N1 c$ @$ n3 h, w8 }Abelian Group of order 1 ( }9 ^2 E4 P- z- `' U' EMapping from: Abelian Group of order 1 to Set of ideals of M . B6 ~1 e) S* }5 b; e1 ^Abelian Group of order 1. ^) M1 ]) A5 Y
Mapping from: Abelian Group of order 1 to Set of ideals of M 6 ]2 ^: X- N7 N! K1 $ j c& H8 \0 l( A: v9 ^1" _+ v# V& [% s) L Q# h
Abelian Group of order 1 5 T7 i$ t; o1 YMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no* [& W) l# X3 V
inverse] 7 V& o# w% b( q% d* W3 k10 s' [+ L- C9 k
Abelian Group of order 1- L: X% S5 L' S9 N) e
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant" r+ W* c( D, _* }
8 given by a rule [no inverse]% m) d' X, d6 S- Y
Abelian Group of order 1 3 Z7 Y' w0 F/ s5 Z% `3 P! _7 ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant7 p! G @: a: p$ J1 F4 d
8 given by a rule [no inverse] + T( U* f! C" c, x( Ntrue [ 5*Q5.1 + 10 ] . K# e; B: ]' W4 \" R- [true [ -5*$.2 ]
lilianjie 发表于 2012-1-9 20:44 0 h8 s& t, ^! D; J, @$ y$ ^1 u7 ]( G. q, A( E X1 L* e
分圆多项式总是原多项式因子：; t7 I X! g0 e
C:=CyclotomicField(5);C;. S& z1 s# q+ F1 Y7 {7 x" s
CyclotomicPolynomial(5);
# @( V' H9 v& }; O2 E" g5 BQ<x> := QuadraticField(8);Q;- M, E( Z0 U( e. \1 l3 z6 f. V
C:=CyclotomicField(8);C;3 R/ k' n# f$ r! z/ u0 n' N
FF:=CyclotomicPolynomial(8);FF;; t" E) _4 e9 e, F
* O$ `+ J# g4 S: ^F := QuadraticField(8);4 T5 z. }& [) F8 C8 Q8 P' B Y$ ~
F; 3 B- U# Q' `- a% ]6 l/ M: mD:=Factorization(FF) ;D; 6 U! a; }( A5 K: Y. cQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field * [: s) D7 s" I' s9 \Cyclotomic Field of order 8 and degree 4" K) b8 @' q1 U! d
$.1^4 + 1 , w; _# G8 O0 E( Y IQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field ) Q) h# G: }, t7 C5 V[ / r$ N+ y* s9 v' M. ~+ ^& V( I <$.1^4 + 1, 1> - G, r9 _# D. r: V# C5 j0 R6 ]1 |] # R) B: R1 ?- [7 s( Y( a& x G7 i0 ?0 j' v% e+ yR.<x> = QQ[] 9 a. H% f- Y7 l" L; |$ m$ S; w* eF6 = factor(x^6 - 1)7 Z4 @! l" W- e0 p7 e7 _& ~, a
F6) Y5 g! G: d% g; `
4 S% F& q9 u& \* }+ M(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) B4 o5 d- Y8 ] 8 E9 N9 G! B" I2 b5 S& Y- x4 W/ JQ<x> := QuadraticField(6);Q; ' v5 Q& r1 q1 m) c2 s( D8 t) CC:=CyclotomicField(6);C; 2 q9 L: E. V" D* \' b* ?. CFF:=CyclotomicPolynomial(6);FF; 8 s g7 ~; B; A* {& k" C0 `% ^ 6 L3 R" L0 ^/ b& g& [' `F := QuadraticField(6); : S! F) U3 u5 D8 a4 BF;1 F2 \3 |: r' I1 e2 |% \$ ]6 }" p
D:=Factorization(FF) ;D;5 c- ?# R9 O# k% v8 t
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field 5 j" ]' j m& {6 YCyclotomic Field of order 6 and degree 2* P3 T) p% G! o- Y/ D9 ]
$.1^2 - $.1 + 12 g1 L# `& P8 Y) Q$ b0 T
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field- j! v3 \7 K' b: E7 K
[% f6 e7 z: q' j$ Y o4 v
<$.1^2 - $.1 + 1, 1># P/ x$ ]; _) K4 R9 Q+ N
]$ I/ B% t$ l. q
3 H- a4 V1 h4 x" N0 e) x
R.<x> = QQ[]; [2 w- y, _: c4 \/ u" Q$ f
F5 = factor(x^10 - 1) ' K" ? x0 m- T4 N" dF5 F) U ~( w3 }0 W, j& n8 {: k( U
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +. e2 X! L) r. A) j- Y. y6 ?- P' H
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1): B+ j/ a1 z* |& E
2 g& P) ~/ b7 V. k2 Y: J1 H$ ?Q<x> := QuadraticField(10);Q; 6 j M: i& E3 jC:=CyclotomicField(10);C;/ k6 G, Q, n, i. Z) X
FF:=CyclotomicPolynomial(10);FF;) S2 G/ Q, @$ i# F+ p& ^6 b
$ Z7 s D; }- d" D* ^9 HF := QuadraticField(10);! H- P/ V6 K1 g# j- v! v
F;! J' D( l6 K6 T/ {
D:=Factorization(FF) ;D; " v& i1 ]; i. ]2 R3 E4 GQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field 8 ^) f" p# |7 Q& o3 E4 GCyclotomic Field of order 10 and degree 4 % C+ L5 L- I9 s) ]( g$.1^4 - $.1^3 + $.1^2 - $.1 + 1 # T! g0 Y7 W8 I* @" h3 ]3 r7 Z f QQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field/ X9 N9 Q& t# K7 M* K( W+ ?8 A% x
[& Z( p8 b; e! R
<$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1> 1 z- Z2 `4 Y- \9 []
IP卡
狗仔卡
发表于 2012-1-4 14:05
转播
淘帖
分享
收藏
支持
反对
微信
提升卡
置顶卡
沉默卡
喧嚣卡
变色卡
抢沙发
显身卡
