本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 1 s1 n: I7 ?* m9 j$ s+ a4 N' d6 ]: V1 u9 [5 g
Q5:=QuadraticField(5) ;5 c& I& a I/ S
Q5;0 s9 M: A9 {' W; @7 g, @
Q<w> :=PolynomialRing(Q5);Q;5 t* M: h# |% H+ c8 B
" }5 S; W N2 b; v" E% \EquationOrder(Q5);. u+ U0 l |* `# l+ g
M:=MaximalOrder(Q5) ;0 {0 q2 ~3 S ~( E, I& `! T
M;* M" H: ~$ `1 A+ Z* n/ @
NumberField(M);6 n& z. F5 D2 r" k/ H
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;% i& `; |4 P: i+ G
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);2 r+ k7 R" z0 n% v
Factorization(w^2-3); - d/ d2 i" s+ d6 yDiscriminant(Q5) ;* g5 }" ]' j0 u+ C8 r* T
FundamentalUnit(Q5) ; , G5 z/ e8 ~9 W# s8 j/ D) XFundamentalUnit(M); : r4 p1 A& p3 K( z5 j% O6 a+ nConductor(Q5) ;0 s9 J9 \) N% g& Z* `' [
Name(Q5, 1);% |/ E( D5 b$ O+ z7 ~
Name(M, 1); - C( u# m/ ^) b& I3 x" ^' `Conductor(M);& S5 m# D0 ^: t; o3 m
ClassGroup(Q5) ; . \: `( J2 a) _5 tClassGroup(M); - r9 X; X# P# e, u, a! `; FClassNumber(Q5) ; ; i! B- A4 `) {ClassNumber(M) ; " P" E: }. z' ], w9 \/ G5 B% S: c5 Q7 |: x) Z$ z
PicardGroup(M) ;) W0 E4 j8 I# D' _; f# C: @
PicardNumber(M) ; & C; s5 o, A' z5 F0 j0 q6 s. p; _7 D" Z+ B- p% L5 V4 T
( H: N) ~+ F8 }% J0 P
QuadraticClassGroupTwoPart(Q5); & u8 a) ?8 Z- c) J9 O; JQuadraticClassGroupTwoPart(M); " i9 x5 ^4 n3 C# x' h1 ^2 E Z9 a- m1 q) T1 }- P t 2 Y- q+ ]) I' a$ u; C% h4 cNormEquation(Q5, 5) ; / A! k; v( ~; l% kNormEquation(M, 5) ;: n2 J0 `1 R% ^0 w! Q* d, T4 Z" G
r( w7 ]3 p" X. {* S ?
; k+ R% j0 M! w6 `0 LQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field 7 ]/ d8 P( `4 T9 i' L# aUnivariate Polynomial Ring in w over Q5: ^4 n* [% Q# ~: l+ X! Q3 ?0 i+ Z
Equation Order of conductor 2 in Q5: y2 a0 g3 T/ }: h* J: i2 g6 J: ^
Maximal Order of Q5 % t- W& z$ j0 D8 |- ^Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field) ?, W" J3 t5 c
Order of conductor 625888888 in Q5 0 X2 R) m5 H( B7 h/ \true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field' \$ ?* g% ^4 ]5 J! O
true Maximal Order of Q53 H9 a% f3 Q3 |' A% v4 ]
true Order of conductor 16 in Q5 ( e3 u6 R0 X, r- G4 ltrue Order of conductor 625 in Q5 + D+ h8 f8 s g! q# e ^6 gtrue Order of conductor 391736900121876544 in Q53 [) h# l n3 V6 T
[ / L$ b% D1 |& a. w <w^2 - 3, 1>8 i; K. H3 S; |
]( i2 U$ H: u& G
5; v- j# k, S- `$ O2 Q# Z: A5 x
1/2*(-Q5.1 + 1)' ~* |) P; e4 ~. M$ O& S# `
-$.2 + 1 . a- \/ [9 ?" o8 i. }, D5 ) {. v( }9 K# J1 e( g, m' pQ5.1, ^% P/ ?* \/ X: R7 @
$.2- ^2 z4 L% f4 x: N3 \' ]
1 6 }7 ?$ a$ W4 M; p9 Y ~Abelian Group of order 1 ! n. F4 T1 i4 R+ f0 O- VMapping from: Abelian Group of order 1 to Set of ideals of M ) ?; K9 P4 A* MAbelian Group of order 12 g( }) D8 F# Z. `& S
Mapping from: Abelian Group of order 1 to Set of ideals of M& H( N. u, U b# a
1) Y0 B" E+ |6 S6 k- D
1 ' t: {2 m) S; d. w8 f, sAbelian Group of order 1! @! W- X" a$ L3 N4 O
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no ! \8 T+ V8 z. i* F1 y9 J# _; ^6 kinverse]: J1 i$ Y( Y+ V. v
11 _) u0 _6 S0 j5 ?
Abelian Group of order 1; d" O8 t5 r Y
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant: m1 g( k$ M0 d1 F) g
5 given by a rule [no inverse]8 y8 x2 U$ E, a8 s% b9 |3 M5 `% \
Abelian Group of order 1 0 G: `& S2 Q( } |2 X4 ?, OMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ^: l3 `6 t7 F" L0 r7 L
5 given by a rule [no inverse] 5 i, F0 \4 F0 O: n! R: X8 Q( m2 ktrue [ 1/2*(Q5.1 + 5) ] 3 k$ q/ }( J& j' f+ f3 R X- Qtrue [ -2*$.2 + 1 ] ) ?; |$ X5 ]/ ?8 e: _# S! U# i8 v' z" l
0 b2 k, r* D0 b: s5 L) C d2 A& F+ d0 }6 H' v }" f D9 ~- @5 ]8 D8 ~
7 m1 l# z. n6 w K1 V. r4 f& d T' t 2 [4 ?9 b& S/ m) [) |+ X8 o" U4 x% Z
4 ^2 t& O- h/ r
==============3 ~, T; ]/ b5 t3 m7 G" E
" G% ]. J9 K+ c" j
Q5:=QuadraticField(50) ;5 y* N6 Y9 y" }$ x0 u# g4 i
Q5; % H3 ?3 J$ p3 s7 m; d( }) H ( Z. y Y- ?0 m1 @# Q) n- bQ<w> :=PolynomialRing(Q5);Q;* E5 }7 |1 _* J
EquationOrder(Q5); % l, c( a+ a- D& i( IM:=MaximalOrder(Q5) ;' P9 G( A* y7 l: ?
M;7 k: u, b8 n$ v x8 A8 S
NumberField(M);$ [' A. M5 U3 @; g. n% I1 g
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;. h1 p) a5 b! e* J4 O! M, L. }) P# H0 H
IsQuadratic(Q5);/ Z& G- N4 e5 ^: R0 l' ?
IsQuadratic(S1); 8 |9 g5 D$ m2 ~3 m d7 cIsQuadratic(S4); ' f2 t. e9 u4 b$ X% f% f" |8 zIsQuadratic(S25); 8 s# P* v5 H* S+ F% r1 |; B) t6 h- hIsQuadratic(S625888888);& P6 o+ l5 x4 J2 o2 G9 T' R( V6 N" V# M
Factorization(w^2-50); & A! j) I4 a! {9 s$ [4 O' qDiscriminant(Q5) ;- ^; o" U) w! [4 g: X
FundamentalUnit(Q5) ;8 l0 H* H' I% \% X9 A, u% p
FundamentalUnit(M); 0 }; V" q- Q& ~Conductor(Q5) ;4 ~. O5 d3 c6 q
! Y) Q$ b) J& B* F5 ]: F+ t
Name(M, 50); " _& i) Q9 R: h4 D$ c* J% A8 |5 b) c* t' hConductor(M); $ G0 T. H1 d7 q+ C, z4 n8 O3 {* kClassGroup(Q5) ; 5 u' E/ G0 F! D! r: V# @( jClassGroup(M);6 s7 M' I* ^4 H5 H3 P/ c- m
ClassNumber(Q5) ;% }/ b) b' c, L% c- ~4 z: B; L
ClassNumber(M) ; & |; ~* y: b9 y/ K+ wPicardGroup(M) ;" v7 ]$ q9 \* B
PicardNumber(M) ;$ \% A3 o v( H* z2 U7 s7 z- D$ d
5 L. \, N+ D5 B y. ?
QuadraticClassGroupTwoPart(Q5); ! _! x# i; l# Z3 t1 U2 o- b) @QuadraticClassGroupTwoPart(M); o- t" N7 U& E# ^8 B t6 J& q2 Q
NormEquation(Q5, 50) ;6 F* n! d, y) F
NormEquation(M, 50) ; ; u( y7 z5 Q! r- E% \ { # p) ?! f# I- rQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field : x" r v7 M% q. ^) SUnivariate Polynomial Ring in w over Q5 5 y/ P- i% ~% zEquation Order of conductor 1 in Q54 j. ]3 W$ R$ Z' I+ Y
Maximal Equation Order of Q5, v3 J- m/ n7 [/ b' B
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field " \) ?# d$ A5 {& DOrder of conductor 625888888 in Q5+ ^$ R$ M, m( M) m; U
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 8 P3 h7 t$ H. n* z; K( Y. R, htrue Maximal Equation Order of Q5& v* t7 | E) {! r8 x9 m
true Order of conductor 1 in Q5 3 d! K y# e, T8 y. Atrue Order of conductor 1 in Q5 7 p9 R9 @0 `3 J, {true Order of conductor 1 in Q55 o( }2 P7 ?' G+ e- q2 ]
[ 5 x, i3 a) g, d' q+ ~! g) s <w - 5*Q5.1, 1>, 6 N6 e* Y: l* k/ @) E, r% e( B5 Q <w + 5*Q5.1, 1> - e0 V" _ ]- ~$ n] $ }* t8 G& o5 A) n- n5 H9 m8. D- E5 g; P+ M( V9 p# O
Q5.1 + 1 3 O6 S# F" j6 J5 T( T. a$.2 + 1 * P' U# V0 f/ Y' \6 L8 - [1 p4 d; q3 ]! t I ' k- U) o0 ?: j>> Name(M, 50); # _: H) g3 @% W" G" c2 ` ^ 5 W1 ?$ t- S. {3 v5 l, MRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]2 j2 c+ k8 \# P' k
; N1 M7 z2 \6 }, _: m" |' H- F8 w$ j
1 * W$ q* }8 H$ @; jAbelian Group of order 19 F$ P. D9 z; i2 v# @
Mapping from: Abelian Group of order 1 to Set of ideals of M 2 i3 c; t) W6 cAbelian Group of order 1 ! |2 A0 k2 I# X S( Y5 \( BMapping from: Abelian Group of order 1 to Set of ideals of M , s5 N2 C0 E2 e6 G s1( z1 A. o# t, W q
18 W8 }) G+ m) e$ |
Abelian Group of order 1 ' G4 d; u+ _3 V1 i" N& W4 qMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no & Z4 i; V' W, rinverse] , S8 M* e& g& w3 ~: Q9 u1% T. r* ~) M5 H/ R' w) \- `8 o( ^
Abelian Group of order 1& R4 R6 E" Y5 I) _' M, t
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant6 s' R. s% a! A* V4 O: B, N
8 given by a rule [no inverse] 8 P' Z; n; ?' `Abelian Group of order 1) [+ E* X1 V- y% m& o3 b4 q9 S
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant: k C" H# k9 g
8 given by a rule [no inverse] % T, R( F6 u. U b0 d) M& [true [ 5*Q5.1 + 10 ]3 R" M ^% z* y# L# R9 X! k
true [ -5*$.2 ]
$ Y3 n8 H% s' ZQ<x> := QuadraticField(8);Q;( I2 _( H7 F0 G; b( T
C:=CyclotomicField(8);C;2 v# c0 x. e" \. w5 c' s9 |, n, b/ I2 ]
FF:=CyclotomicPolynomial(8);FF; 5 u$ A, R$ ~/ K7 w, _& e; F. n" D' L5 Z" a
F := QuadraticField(8); . h! K% J# V7 B1 \% v* w1 aF; 8 C! j4 v5 c6 U4 G5 eD:=Factorization(FF) ;D; ( M& e3 Y% d4 ]+ p( M# D7 W9 gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field a& ?4 F. S* p8 E4 a
Cyclotomic Field of order 8 and degree 4! i# ^- b T* M; m4 Y
$.1^4 + 1) k3 P$ H$ j5 `+ \0 e
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field1 H. m% r- B9 I0 \5 D: |, w
[ - `: O- ^" K6 D* H @ <$.1^4 + 1, 1>: N J$ L" v0 f+ ?' a
]; X R5 n! c5 b. B
, i8 W& L& S; P/ j; wR.<x> = QQ[]! ] B3 e6 z" \! ~
F6 = factor(x^6 - 1)" [* s% \( V+ X2 J; ]2 K
F6 5 Y3 k' ^" q) E* Z- k . t5 W) L. o, a# ~9 [3 _! {(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 a; Q" G9 N$ b; E0 i5 @5 n: w$ y% A
+ ? j" L- }- e% n) C9 l$ }/ E/ `
Q<x> := QuadraticField(6);Q; 6 K2 d- ^, X+ U( _2 fC:=CyclotomicField(6);C; & x) c! M0 Y+ |% h! F qFF:=CyclotomicPolynomial(6);FF; 7 y. z: b# a* z- v1 L n6 G5 A" |1 z# T1 [* H4 w
F := QuadraticField(6); 2 z" T- F1 e# Z: H1 b: aF; : e F+ D! B9 O( H9 ~: u( SD:=Factorization(FF) ;D;( B; W: k9 V+ v8 X% h7 V( u) O
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field! H" ~" M: T* v& z
Cyclotomic Field of order 6 and degree 2* J/ N2 M9 m3 l" M1 d3 Q" [
$.1^2 - $.1 + 1 # l1 a- A* \5 F3 \1 fQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field0 V' g' t& L/ b
[ 4 g( U; S+ ~$ d* d' c' _( z <$.1^2 - $.1 + 1, 1> ! b9 W; [2 c Q3 |9 _] , b2 y" `" O1 U5 [: |3 B: G 8 k4 J$ _& @8 L+ _R.<x> = QQ[]6 ^9 I5 R2 j. g0 I' M" X
F5 = factor(x^10 - 1)0 L9 c7 R- F: [/ @
F5' E" d/ ]6 O+ g3 |$ W
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1 l5 n; [ a, Z0 M/ w* O1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1) ! K5 }5 j6 g/ P- R3 G q( R5 e , y( x2 ~* a% j/ W: TQ<x> := QuadraticField(10);Q;! j+ K& K0 g2 ^+ G7 g; T7 s
C:=CyclotomicField(10);C;- r% l0 e4 Q+ H% q3 h. w: V4 f
FF:=CyclotomicPolynomial(10);FF;2 \6 b- `: V M+ a
9 n- T) G6 q4 m7 H7 V7 OF := QuadraticField(10); ) g' R3 a) k# Y4 }: L. D* `F;% N/ U) E* t+ |5 z$ a; b
D:=Factorization(FF) ;D; 5 J" n( Y" o; P, h5 T; Q) x/ L, _Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field9 _5 E' p z( g+ }7 O9 k
Cyclotomic Field of order 10 and degree 4, x. w* F& H, T0 v5 a
$.1^4 - $.1^3 + $.1^2 - $.1 + 1$ L- B+ q3 v' a$ ]5 E) b
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field$ M3 d9 a8 Y0 N) \
[! p6 c' A3 m+ `/ d5 E4 V2 D
<$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1> ' G0 `3 z3 |$ a6 g* Z8 U]
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发表于 2012-1-4 14:05
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