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虚二次域例两（-5/50）

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lilianjie        

43

主题

4

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204

积分

升级  52%

  • TA的每日心情
    开心
    2012-1-13 11:05

    • 签到天数: 15 天

    [LV.4]偶尔看看III
    跳转到指定楼层
    1#
    发表于 2012-1-4 17:41 |只看该作者 |倒序浏览
    |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 . C' V  z& R. w# }8 k4 ], F
    ; f* J0 }6 H( m1 D
    Q5:=QuadraticField(-5) ;
    7 y* D6 |( Y% R$ t) {Q5;3 |1 t3 o; }2 D/ V1 G" d8 |
    : A% P5 ?* K, D! A
    Q<w> :=PolynomialRing(Q5);Q;
    / l! G9 M6 T& dEquationOrder(Q5);
    3 M8 x% ~; g4 W' kM:=MaximalOrder(Q5) ;
    , ~8 r& m$ ]% X' k. m' U6 G& k3 ~M;3 d! |. V# F& d# x/ F' O- D
    NumberField(M);) W, W4 ^# J# N! C6 g/ R
    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;
    7 }! g: e8 U( o1 w) }) a' LIsQuadratic(Q5);
    . |) a" F& d/ A* b$ G& z' x/ HIsQuadratic(S1);
    " l- J. S8 @) G3 E4 bIsQuadratic(S4);$ ]6 L" M% d. M
    IsQuadratic(S25);3 G% K7 U$ a; Q  I9 W
    IsQuadratic(S625888888);
    $ u' K4 ?3 s1 }$ e0 X+ n$ \  Z' d4 jFactorization(w^2+5);  
    ( N8 z6 J" K( x: ^Discriminant(Q5) ;2 l4 H6 x/ M. }
    FundamentalUnit(Q5) ;
    - P% |3 @5 g1 VFundamentalUnit(M);+ A- r2 B8 X# I& I% ]
    Conductor(Q5) ;
    - @. V" U/ i. X. [1 y) z/ T( j, L9 e' e0 ]  E/ h0 s$ O7 D! x% k9 }
    Name(M, -5);7 D' U& @1 Y# U
    Conductor(M);7 K1 ]& d; ^2 W
    ClassGroup(Q5) ;
    + Q5 Y6 ~# h6 K/ z, e( @$ p; R$ vClassGroup(M);
    0 B" G+ \- d1 R/ ~. ^ClassNumber(Q5) ;
    : X7 g0 t, n( Z6 m4 }' a- NClassNumber(M) ;
    7 ]0 H5 S1 m& ~. `& ^2 wPicardGroup(M) ;8 c! s5 B2 X7 z* a3 q
    PicardNumber(M) ;
    - u! s- x: W" A$ F8 u' P4 ]' E3 S, R4 U  h
    QuadraticClassGroupTwoPart(Q5);
    0 \' x3 U/ u" m3 ?; HQuadraticClassGroupTwoPart(M);
    2 p( T( G6 t& x. P3 n0 e9 bNormEquation(Q5, -5) ;4 n5 P* r5 c& Z( [6 r
    NormEquation(M, -5) ;
    ! F( i- f" T' z% u9 G3 `Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    3 c4 h- m; n5 _4 I% b8 q/ `- ~+ o8 ZUnivariate Polynomial Ring in w over Q5# f; F# i8 C9 P- n5 G% y, A# z
    Equation Order of conductor 1 in Q5
    / S: ~' M- p3 A% G6 z6 MMaximal Equation Order of Q5
    : p7 s# H( C1 cQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    . ]# I1 f/ x8 `5 l5 ^Order of conductor 625888888 in Q5% E. Y- U9 p# V+ J8 [6 ^
    true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    0 @2 w+ d8 z! vtrue Maximal Equation Order of Q5
    % @0 o. d6 L7 u, w/ ^4 g5 Utrue Order of conductor 1 in Q5
    % j4 N1 D" y5 [- E4 otrue Order of conductor 1 in Q5$ X- r/ s; X4 M0 w8 t
    true Order of conductor 1 in Q5
    3 N8 A4 _' d. x4 B# z8 c6 Q) {. F[# a: U9 f! b4 D8 `% s
        <w - Q5.1, 1>,
    0 ?/ |: y- Q7 f  a    <w + Q5.1, 1>' q5 a& y6 L; D1 d1 i" i
    ]$ w; L8 V5 `8 V6 m+ t! v
    -20
    ! I9 g' i$ a4 s8 T8 k1 v! d
    , `% F# _" d* v, j>> FundamentalUnit(Q5) ;0 H% e! u* q  o& |% ~
                      ^
    ; x; f* v8 K  aRuntime error in 'FundamentalUnit': Field must have positive discriminant! }3 Y" e* P6 C7 l6 Z* w0 W

    2 A6 r6 ~2 A; k% j
    2 Q! p& f1 X( B6 o>> FundamentalUnit(M);# `  b; ^5 q; S. s& L
                      ^0 t1 \( S  o9 f+ @  Y5 U
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    0 R' R; F" X7 v, Q7 k
    0 K) p- W( z, U/ j20
    4 e+ r. D! \1 p" }+ I
    5 d0 N; B* k% h! C' n. x>> Name(M, -5);) W& F2 q0 z# h
           ^; f3 S& R" b' W* Z+ `
    Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
    + Q* O. w- ]$ d
    1 I2 m( x2 I3 n5 A& ]: f7 K, B10 y- ~5 n- v7 M# _
    Abelian Group isomorphic to Z/2
    ) Y$ M- T, U  @/ k+ C; t+ W, v" Z# EDefined on 1 generator; L& L7 u; [; u- Y
    Relations:
      U& E+ G4 E! s5 P    2*$.1 = 0% m; _( U# @# e) Q& A+ R( E: p
    Mapping from: Abelian Group isomorphic to Z/23 U& \8 k! [- }# O- M; g$ x5 {
    Defined on 1 generator6 U8 y  W) G9 u$ C' M
    Relations:
    6 m; {6 }, |$ O# v    2*$.1 = 0 to Set of ideals of M
    * c& L0 v; k4 _Abelian Group isomorphic to Z/2
    3 D6 W; Q8 ]7 jDefined on 1 generator
    7 X/ U: ^2 |& j( `" g# |Relations:
    1 X' d$ A3 T" x    2*$.1 = 0
    % a; u  t8 a2 e2 S3 IMapping from: Abelian Group isomorphic to Z/2
    ' Y8 H' F; o  b/ D6 ~8 @Defined on 1 generator! t; T5 e9 X- \8 N, D8 L
    Relations:2 s+ }0 n6 s4 f8 V# v
        2*$.1 = 0 to Set of ideals of M
    % g/ }' V: h! C0 j: I% R4 `$ v- f2
    % X7 f  z. J1 J. n  Y; W) w2
    9 G4 B0 A: k1 A% D7 DAbelian Group isomorphic to Z/2/ m: f1 r' t+ t
    Defined on 1 generator$ k  Y2 B# Q6 v
    Relations:# E9 L, Y+ _, S6 V4 ]5 R
        2*$.1 = 0
    5 O" o1 W0 H7 Q6 XMapping from: Abelian Group isomorphic to Z/2+ _# J0 W& G( z: B2 x
    Defined on 1 generator, s, X! d/ O/ P* Y% C% ~: L
    Relations:% F- s4 U% R: w7 q6 v0 `& e* n
        2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]  v$ B( a; [+ G, X1 v3 c9 {
    2
    & v. x2 R& c9 \Abelian Group isomorphic to Z/2
    # t0 k7 i0 B1 p5 KDefined on 1 generator5 Y9 a7 O' o0 Q1 v
    Relations:
    - l( D' C4 X" p7 Y  S    2*$.1 = 0
    % y. N  F" m( g" U2 VMapping from: Abelian Group isomorphic to Z/2$ B4 J! F: `. ^" V- z
    Defined on 1 generator) ~' z% b2 _. M6 x2 \5 p
    Relations:
      I& }0 w9 h+ {6 ~2 W    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no 9 D' }+ r7 J" c  W  j! r0 o0 o
    inverse]
    & i7 F9 h. E+ C! ^! iAbelian Group isomorphic to Z/2
    " J0 L2 ^6 v$ u# y+ @Defined on 1 generator
    # O5 |$ b. y/ i/ `4 HRelations:
    ) z% @/ o+ ~. {7 ?/ N) S    2*$.1 = 0. H, ~6 k7 @/ H3 ~: F7 i: j
    Mapping from: Abelian Group isomorphic to Z/2
    * f" A3 h. [" F$ b. D9 G3 SDefined on 1 generator6 C% j) K; b' u6 w( q# G8 K  g
    Relations:
    & p7 |$ ^  F5 c! o$ w0 F: o    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
    4 V% }' Y9 a1 i; pinverse]
    3 n8 o7 Z7 u8 m$ H: |false
    + ~- A. k6 c( g! ?: bfalse
    # o" z$ N0 D$ s6 D+ U1 H( M==============3 a9 }5 k0 O( {1 C

    , S+ \; }3 d, S; Y) A' N. P0 a5 |& v. F0 ^* L# }& e
    Q5:=QuadraticField(-50) ;3 ]8 r9 W' d& W- Q1 s/ M, M+ K0 I
    Q5;
    9 C2 |, |9 ?; O: m* L" e& x2 T+ f9 Q4 z" {; R& \1 g
    Q<w> :=PolynomialRing(Q5);Q;
    & h; B+ A  J9 P" B* N# k% o' jEquationOrder(Q5);. S5 M, n6 T" n# ~
    M:=MaximalOrder(Q5) ;2 B! j- H" r/ V2 {3 @
    M;% c6 V" n' ~0 C1 e+ ~
    NumberField(M);) L. C& h$ U" H0 Y% P
    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;
    + h; O/ L8 l- R. sIsQuadratic(Q5);
    - N2 u, }  ]% W9 }IsQuadratic(S1);. Y% q! B4 f1 A; f3 F
    IsQuadratic(S4);
    + f% B; |  H2 P3 Q3 C6 ~+ M- A% D$ _IsQuadratic(S25);$ h0 d/ V. D" W' o8 X* X
    IsQuadratic(S625888888);! A& {8 u' X5 e: C. n8 V1 S
    Factorization(w^2+50);  
    ! z4 p/ a1 Q2 }7 j8 K" [Discriminant(Q5) ;9 G0 @9 A/ G+ }9 ?
    FundamentalUnit(Q5) ;
    7 [! c% R7 G( X: T" \6 z4 BFundamentalUnit(M);
    / m$ l( ^. n4 K/ }+ L1 @# B! R# xConductor(Q5) ;
    , L5 K. a% W+ C/ M7 N3 `, V5 P  s6 ?1 s. c3 a
    Name(M, -50);: w5 Z* ]1 D2 P2 M2 f
    Conductor(M);
    - [  m( _! t! k2 i) p2 s- FClassGroup(Q5) ;
    4 x+ Z5 R" o3 X. M8 vClassGroup(M);7 q7 q' n2 o% f# ^! w- n3 z! x  v
    ClassNumber(Q5) ;
    # d. z9 {7 b. i+ f: MClassNumber(M) ;! _; q' A7 |; y6 k8 W' J8 m- q
    PicardGroup(M) ;
    " \8 p  p5 b5 A' N' ^PicardNumber(M) ;* w: t# l* [) D1 w0 \+ t
    2 [  U) E; v" X0 J
    QuadraticClassGroupTwoPart(Q5);
    8 J2 i3 @; \  B4 j: X% kQuadraticClassGroupTwoPart(M);
    . R! _! W6 e4 h* X7 V/ z& `" X8 iNormEquation(Q5, -50) ;3 V: V: o+ C% d' n8 N
    NormEquation(M, -50) ;& P0 s$ p5 N* m& y) R" d7 G  N2 T: V
    7 F& V2 k5 A; D2 o
    Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field* T" I' P$ C  h% l6 Q& ~
    Univariate Polynomial Ring in w over Q5- \& w) ?" c6 i6 U8 a* u9 Z
    Equation Order of conductor 1 in Q5' ~! D6 ?! G' Q, e! y
    Maximal Equation Order of Q5
    9 }% F7 P6 J) v1 T1 [* lQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field7 }0 b3 h9 i& u2 d: a5 u- i
    Order of conductor 625888888 in Q5/ m9 b( I+ }6 [
    true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field7 B9 c; y: H5 H/ r/ f3 t
    true Maximal Equation Order of Q5
    ; s6 `# X1 O- {true Order of conductor 1 in Q56 x6 b9 N+ L0 ^) g, ~
    true Order of conductor 1 in Q5% h9 z5 u) y% V/ ~& ]- s
    true Order of conductor 1 in Q5, P' @6 s9 _3 Z+ l, b1 b1 m
    [
    7 C: {( v5 s  _0 T    <w - 5*Q5.1, 1>,
    # b6 }5 U, W" G4 e% x0 O    <w + 5*Q5.1, 1>$ g1 q# U! {0 J7 T8 W
    ]' @, k: H5 G9 H- H1 e$ K2 f
    -8
    , n( b9 j6 r( Q# i% k
    1 N$ T: q+ G/ `# v) ~' @>> FundamentalUnit(Q5) ;
    / ~; F4 T5 f. S5 h! E! y                  ^
    . v3 i, J+ v0 a* f9 i+ ZRuntime error in 'FundamentalUnit': Field must have positive discriminant
    0 Z. b, Z6 Y+ o* p! x4 x9 h- Z( |. o) f% [

    - h/ [/ m1 ^& d; J) M/ ~>> FundamentalUnit(M);% m' N% w# s( l. `6 C: A+ Z
                      ^
    ( h8 z+ `' A. ?Runtime error in 'FundamentalUnit': Field must have positive discriminant6 |* Q. W+ w$ s

    $ b" P' e. U4 A) V83 A9 K; \! A/ g; B! F, \, O1 \% ^
    9 }# H" W# {" |6 C* r
    >> Name(M, -50);
    4 V- `/ k! |0 k- ~/ Z9 n+ J! i       ^
    * u) I$ q1 l  r6 _- V  ZRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]9 J( F0 I' i: c+ X$ o# I" @, a
    $ o$ \" J, F* C
    1( \# U( X% q) e% [' N' k" a: `
    Abelian Group of order 1
    ( ~7 L+ Q5 l) l0 g$ s1 ~2 hMapping from: Abelian Group of order 1 to Set of ideals of M
    7 ~' w" g; }" q+ I2 gAbelian Group of order 12 r: k- }' X1 n8 `
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    + r7 ^! u1 E- G0 t1) _! d& h! o" O* W+ S( n7 V
    1# `( q  B0 ^" ~' U* d
    Abelian Group of order 1
    # c" ^$ n& P. I7 T# PMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    * K- b3 _0 p& z) e) tinverse]2 v. M$ L$ [# q; ]. {, [# @! A# J
    1* r# e$ k. e! X  j* C1 @
    Abelian Group of order 1: l+ I! |' G) I. _8 ?" t6 X2 k
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    5 s! X, u  [/ o0 a9 A" |. g. ~-8 given by a rule [no inverse]
    # Z# Z+ ~9 j; O. C# xAbelian Group of order 1. l, v' l# @2 c8 Y" C5 {: Z
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
      b% y2 y/ T9 m7 a( `. G-8 given by a rule [no inverse]
    ; Q6 I+ T# r* a9 u+ v& n" @false
    * {5 j* B2 Z# Q! Z+ Hfalse$ t1 K- L0 {: m" ]  u. k. A8 S
    zan
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    lilianjie        

    43

    主题

    4

    听众

    204

    积分

    升级  52%

  • TA的每日心情
    开心
    2012-1-13 11:05

    • 签到天数: 15 天

    [LV.4]偶尔看看III
    2#
    发表于 2012-1-4 17:51 |只看该作者 |招呼Ta 关注Ta
    看看-1.-3的两种：
    " W+ W% |' T% x9 {% X' O$ ^; F' W/ X0 ~5 `6 K+ X! B
    Q5:=QuadraticField(-1) ;0 e4 c( Q! l8 L( o# ?4 K
    Q5;- c' ^. Z" z' Q% {
    0 {3 D! R1 v# s* _3 z, w+ a' i' q
    Q<w> :=PolynomialRing(Q5);Q;
    0 w" Z" r5 Z7 L- M& |- K7 iEquationOrder(Q5);
    : N0 d! T( z& ?+ h8 W' gM:=MaximalOrder(Q5) ;
    8 Q* o) H# C1 d+ ~M;. d# F' P* `' ^# m$ y& {
    NumberField(M);
    + K. ~2 ]$ r3 A# }, 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;: P  Q; p+ I4 M! |; r6 k9 @
    IsQuadratic(Q5);+ Q# d# p$ q) W
    IsQuadratic(S1);( f+ g' X6 L" l; g; i( X
    IsQuadratic(S4);
    0 x3 K3 N/ Z8 I+ {IsQuadratic(S25);# l8 o/ q8 C) p
    IsQuadratic(S625888888);; l+ d& z( b; o
    Factorization(w^2+1);  
    % [5 _9 L" Z! ^# qDiscriminant(Q5) ;/ J  q, {  y$ K
    FundamentalUnit(Q5) ;
    : O0 }) E. E' ~3 O4 B' @FundamentalUnit(M);
    ' o2 f  m- V8 Z( f( mConductor(Q5) ;, L! c% d- z8 B; }% x

    0 q8 w9 Y, h5 A4 uName(M, -1);* p, }2 x" v- d; I1 z) E- Y
    Conductor(M);, b$ c# W0 U) ?* c' D' b: ^
    ClassGroup(Q5) ; 5 l0 D& I6 j! B7 g- e% }
    ClassGroup(M);
    # u1 f3 M) H3 A, k, N! J% w* dClassNumber(Q5) ;5 e( }0 }# b" d8 u" H; @4 T* [
    ClassNumber(M) ;
      M4 D, o4 x( `; y- p/ M( BPicardGroup(M) ;* f* V( I2 w, W! T3 C
    PicardNumber(M) ;
    9 `0 T. i; y6 }# ]& B! v0 W. l/ N8 }! y! o4 B3 H% y, O3 m
    QuadraticClassGroupTwoPart(Q5);: Z( R1 N( ~$ a
    QuadraticClassGroupTwoPart(M);
    2 f5 q: s( N: @8 u4 [; l. D$ T6 gNormEquation(Q5, -1) ;
    2 i# Q. V2 y8 ~9 F/ k& m: w2 O( ONormEquation(M, -1) ;
    , y3 `: [6 h/ W8 e, E4 }8 I" F3 r' Z3 N" j
    Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    ; I/ P8 q9 Z: a# iUnivariate Polynomial Ring in w over Q5
    9 Q, G% L! Y7 b' aEquation Order of conductor 1 in Q50 R1 `- _' C! g5 I5 v
    Maximal Equation Order of Q5' U( e9 K. U, Y& c
    Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field& S. v1 @; b" N, y
    Order of conductor 625888888 in Q53 x# n/ O' [: u* i4 }
    true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field5 T' Z1 G% e& s% Z; h
    true Maximal Equation Order of Q5" Q5 A% p0 w8 B7 x
    true Order of conductor 1 in Q5' C+ C) f0 J3 N4 `! a' y, {
    true Order of conductor 1 in Q5$ W: \+ O3 R& G7 c! @. h
    true Order of conductor 1 in Q51 F4 v6 K, {1 ]8 l* {7 A
    [$ {2 d' J$ d' M9 O
        <w - Q5.1, 1>,
    9 [/ @  c  K5 m- d4 }5 p    <w + Q5.1, 1>
    7 {7 e/ i) g! b2 H9 ?]
    1 A+ \' o! `; T, n* {  r-4& x( Z( J+ O5 U. Z7 ?2 e& c0 m" i
    5 w* k9 p( e7 ^' }8 g6 b) m
    >> FundamentalUnit(Q5) ;
    4 A* y+ V* t$ U, i7 P2 ~                  ^
    1 x5 b3 U% N: E" |( \$ Y4 |3 V. f% fRuntime error in 'FundamentalUnit': Field must have positive discriminant
    9 q, `" k% g" K. H( H* i- {; l0 ~/ v2 I

    * q. G5 p- L6 ~8 `>> FundamentalUnit(M);8 \9 D0 a7 c1 J3 f' T
                      ^
    ' I6 }# J9 B) W( X' CRuntime error in 'FundamentalUnit': Field must have positive discriminant
      s$ ?5 E, q: I7 b) c1 ?2 O! ^0 L' n  v5 l5 e* V* R
    4
    & t5 D( Z- x, b# G9 z) y4 F6 B  C( D8 Y; Q  q7 X
    >> Name(M, -1);& J4 x) ~# b$ @: n
           ^
    * E: B8 x; ~" QRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]7 {. k9 F" ~% W7 C: o3 O. T. B
    4 s6 S" u0 f# B% r! _' L
    1
    6 r3 x, Y6 v7 mAbelian Group of order 1
    $ H4 c0 u9 z% y4 YMapping from: Abelian Group of order 1 to Set of ideals of M
    0 c4 e: P+ C9 v) }& B$ c* h. _  }Abelian Group of order 1
    ) ^4 m% w3 H/ K  o; ^% y$ N. w9 `* ]Mapping from: Abelian Group of order 1 to Set of ideals of M
    ' |6 ]8 V9 x; [3 b4 i) v1
    % E  ~: o% K+ k& R1
    5 x- ]# T3 s. z/ O$ L6 XAbelian Group of order 14 M0 U$ |# ~7 n3 M3 N2 g
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no4 f0 Y1 y. i; F& U2 g8 x
    inverse]
    5 t. j) i% I" M, Q0 O1. i2 m; S+ w; T5 y* J+ F
    Abelian Group of order 10 h+ _+ Q  @! b' x/ a8 l% |5 Z
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    . p: d$ V' G1 E" c$ S-4 given by a rule [no inverse]
      X% m9 h8 i" L; BAbelian Group of order 1. Q: d$ v' Y# ~7 V. t6 [. `! j2 E/ |
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant) f( y* O, p) |/ h3 [' ]
    -4 given by a rule [no inverse]
    ' {' P# @# s  N; w/ D( Bfalse
    7 U: v* W  `/ A! xfalse# Z% U3 b" ^! x1 }2 e3 V
    ===============
    " K6 R! x! b: v. H6 y* ~" i7 O) h. K- @
    Q5:=QuadraticField(-3) ;; e! m( i7 v1 L3 {) D, z! c
    Q5;
    ' b9 t3 T7 q: D; S, d9 I2 H& ~% x2 D9 w: f7 \* g) |0 c
    Q<w> :=PolynomialRing(Q5);Q;, F* C$ Z2 b0 A$ j1 J
    EquationOrder(Q5);6 F. K- l7 h1 e" e: _: [
    M:=MaximalOrder(Q5) ;
    5 p6 ^  U& a' XM;$ g+ e/ G7 r  Z/ X
    NumberField(M);
    5 h' i7 b$ ~0 TS1:=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;
    ; D) _5 z) g+ L" a- qIsQuadratic(Q5);, _' U6 n/ }: ?9 }
    IsQuadratic(S1);
    $ p& |  f  u5 j6 ZIsQuadratic(S4);
    8 G/ P; H/ l- h, I5 o  _3 AIsQuadratic(S25);- Z+ S- t1 p6 V- D; z
    IsQuadratic(S625888888);2 j5 G% M7 M0 y& I- R, v9 p
    Factorization(w^2+3);  $ N* N' j( I: d& ]3 H% J
    Discriminant(Q5) ;
    1 K! F" m' b9 v' G% EFundamentalUnit(Q5) ;
    7 b, N% i# K: a  s4 m$ t7 AFundamentalUnit(M);
    9 n' ~* r0 j+ l" ?8 c" B; nConductor(Q5) ;" S1 w+ Y6 o; l# U4 U6 k, a  f
    $ o, y! s+ e' r3 P& R8 p9 z3 U
    Name(M, -3);
    % G+ H! M' \% Q. [! J1 \$ f* kConductor(M);
    3 H3 c0 X% x, g% A- }) h# i; C3 }- RClassGroup(Q5) ;
    & g" H/ M* r8 y1 @7 \9 nClassGroup(M);# Y" I1 f4 ?1 H' z3 V8 M6 {
    ClassNumber(Q5) ;
    $ u- [" `- F2 j4 p; ?: @8 YClassNumber(M) ;. Z5 _' z  I. `
    PicardGroup(M) ;
    , \# d7 S5 _: G. b& ~PicardNumber(M) ;2 T8 `! F  D+ p. m6 W
    " c- y1 q- B, v# ^# i. I- E+ N4 ~
    QuadraticClassGroupTwoPart(Q5);4 l. p0 e# |! \, H( D" Z
    QuadraticClassGroupTwoPart(M);
    - a& c% L% W  U* D" ANormEquation(Q5, -3) ;
    " _( N) s# u2 W) [' T& V! N$ @NormEquation(M, -3) ;
    3 g5 L5 U4 F2 ~* V) l
    9 W8 S8 U6 O( A9 K0 V$ A' r. jQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field  P  d5 E" q" e3 ]
    Univariate Polynomial Ring in w over Q5# n( v. F2 C% |. E2 L
    Equation Order of conductor 2 in Q50 a7 m  r3 S2 R+ D3 i$ L
    Maximal Order of Q5# b0 a2 u' f2 j2 R' Y+ }
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field% [& B9 j/ ^; k
    Order of conductor 625888888 in Q51 D! y+ C( i5 r! D
    true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    . D4 Z: E- e! M* gtrue Maximal Order of Q5% p2 N1 [7 q, J
    true Order of conductor 16 in Q5
    ( |) d0 C& i, Ttrue Order of conductor 625 in Q5+ D) y* j7 C' X0 ~# @
    true Order of conductor 391736900121876544 in Q55 |! y: y0 |% n; v) W
    [
    + V$ R- A- B( o( k% D! F    <w - Q5.1, 1>,& d  w9 S9 k3 B- d0 O! q& h- M$ \
        <w + Q5.1, 1>" v+ V9 m  O0 t1 T
    ]  B4 ?3 ], w9 ?3 Z
    -37 H- c! a) g4 B' ?: V- v& T. x

    * d/ ?+ _1 S) V" `9 T  a>> FundamentalUnit(Q5) ;$ z8 G" E& G: X# s/ @+ U9 W  \* m
                      ^3 O8 w. k! q# ?0 ^" D, w: Z4 o  U
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    ( z% o+ m6 a/ B8 \9 y, Z
    4 g& Y9 v0 z: n
    ' G0 |$ _8 W5 ^% s* ?>> FundamentalUnit(M);
    . x+ j0 T3 y3 l                  ^8 M, |7 V& U# K; L2 \; G9 L8 k# g
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    $ P: a) t6 P2 U2 z+ z0 l3 s2 t4 I, L$ W) p8 `% V0 D
    3" \% P. u3 U+ O; c, g
    3 U3 D9 ~1 k" [: ^% n2 _( @+ H' {
    >> Name(M, -3);
    ) Q: M" |4 n$ F- \  s6 r       ^
    0 [5 k6 [- L3 \" jRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]- M* U* _  O, ~) Q2 V

    & y# h) N' @7 c/ j4 ^; a5 V7 A' @1
    , w; l$ q4 }# K  S+ \- @Abelian Group of order 13 E) T' k# I# U, H0 Z
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    2 l' l. ?* d* l  h0 {" Q* g/ R0 @Abelian Group of order 1: O- |' O! s5 _0 j+ T, m
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    2 r  Z+ |5 H- E. [0 e) R12 D% Z" u) d7 v9 f9 v+ Q
    16 q+ {) G! O& j$ T
    Abelian Group of order 1# o+ a$ y: t. t5 J. m
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
      Y) {8 }- }9 V+ }$ X0 W. N% |inverse]4 V% {: W8 k" s5 D& F
    1
    6 {& \  @: R4 o! @6 [1 e2 `Abelian Group of order 1
    ; B' r* M" d, l/ L# L5 o+ V/ dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant  [* I. X5 M# H1 b, M7 B; ~
    -3 given by a rule [no inverse]
    - \3 J6 U; e/ X3 i' rAbelian Group of order 1
    ; y5 f9 r) b3 ]8 O7 ]; ^) cMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    9 A4 J! T+ Z  P5 h-3 given by a rule [no inverse]! Y% X8 h" G2 I" D/ L! N4 M
    false) h) @4 |- N8 v/ P( `
    false
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    孤寂冷逍遥 实名认证    中国数模人才认证   

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    [LV.9]以坛为家II

    社区QQ达人 邮箱绑定达人 发帖功臣 最具活力勋章

    群组: 数学建摸协会

    群组: Matlab讨论组

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    群组: 数学建模

    群组: LINGO
    3#
    发表于 2012-1-5 08:36 |只看该作者 |招呼Ta 关注Ta |邮箱已经成功绑定
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    [LV.4]偶尔看看III
    4#
    发表于 2012-1-5 13:02 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-5 15:20 编辑
    : o7 X/ ~6 g! q; M/ i4 Q5 i( v! \" i
    9 [* S7 U) L3 b! D8 {  N2 f) c3 }Dirichlet character
    $ @) c# @* O! \3 CDirichlet class number formula
    7 b8 U4 s0 n2 w# Y( H3 h# }  G: o: ]7 ]. n+ F4 W! C$ f
    虚二次域复点1，实点为0，实二次域复点0，实点为2，实二次域只有两单位根
    9 p; g' L9 d; J7 f
    : j) w, W# _* E1 ?9 [. O8 {-1时，4个单位根1，-1, i, -i,w=4,            N=4，互素（1，3），    (Z/4Z)*------->C*      χ（1mod4）=1，χ（3mod4）=-1，h=4/（2*4）*Σ[1*1+（3*（-1）]=11 [7 T# T: w9 n" I- ~4 Z
    ' J! Y1 e: s5 G* D* b
    -3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
    " t, G4 Q& C0 e1 f# R1 v  C& |# L: J" ~7 ~h=-6/（2*3）*Σ[1*1+（2*（-1）]=1
    - |& i! Q) v; B/ Q9 |
    9 i! T7 A6 U. h7 Z-5时  2个单位根                              N=20   N=3  互素（1，3，7,9,11,13,17,19），    (Z/5Z)*------->C*         χ（1mod20）=1，χ（3mod20）=-1， χ（7mod20）=1， χ（9mod20）=1，χ（11mod20）=1，χ（13mod20）=1，χ（17mod20）=1，χ（19mod20）=1，, x  g0 i, U+ `) h. l1 i" Z

    5 f" t/ W& r* O% r* \+ _# W& R# _* w8 p0 C% Y
    " [3 l* l1 v: e8 @/ Y
    h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
    - b6 Y5 _# s# v' q' t, D# l% Y$ b4 l
    $ ^( |8 R/ p& X* y, }# }; ?' Y  a+ x

    0 Q4 e- W) R$ N) G-50时  个单位根                          N=200
    5 u& `$ H8 e! N1 j3 C5 ~* m
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    [LV.3]偶尔看看II
    5#
    发表于 2012-1-5 13:51 |只看该作者 |招呼Ta 关注Ta
    Dirichlet character

    21.JPG (79.18 KB, 下载次数: 260)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 265)

    11.JPG

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    [LV.3]偶尔看看II
    6#
    发表于 2012-1-5 20:37 |只看该作者 |招呼Ta 关注Ta
    pell   equation

    11.GIF (22.03 KB, 下载次数: 266)

    11.GIF

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    [LV.4]偶尔看看III
    7#
    发表于 2012-1-9 20:28 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-9 20:30 编辑 5 u6 C7 h" l! R; ]+ o7 \

    + H! z1 j  _0 `, Q0 t& eF := QuadraticField(NextPrime(5));% ~0 s# r0 O- a% Z3 F8 q

    6 d* ~% s* d& b" I, QKK := QuadraticField(7);KK;) g) i( C( k  ^9 _! |
    K:=MaximalOrder(KK);
    8 ^* `+ t# T9 R2 I. IConductor(KK);+ w% G+ ~& ~- j
    ClassGroup(KK) ;
    & t" M4 p2 j  `, dQuadraticClassGroupTwoPart(KK) ;
    $ j0 N# T) _4 `# |1 tNormEquation(F, 7);
    $ i7 e* P, ~. q' V5 |4 lA:=K!7;A;
    + U, d: e- z5 ~! t4 SB:=K!14;B;
    " ?& g; Q- w  mDiscriminant(KK)3 [% ]1 |1 Y" T, z' t3 l) N( N

    : m' s4 @) U  d" ]+ D& eQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field% _. E6 D3 r( f
    28; X# i) p6 j/ L
    Abelian Group of order 1( [  Q# {) `! ]( C; T& v& A
    Mapping from: Abelian Group of order 1 to Set of ideals of K. K# ?6 m7 X+ |
    Abelian Group isomorphic to Z/2
    ' @3 t5 D0 v  o& J8 }0 [' O  BDefined on 1 generator& }& ~' u. M1 q$ l9 N
    Relations:
    , V: j8 R3 a' u! D, p  S( W    2*$.1 = 0
    & V6 J- v7 P: [, [: i! h) c1 r" `Mapping from: Abelian Group isomorphic to Z/23 |& }  `, ]+ s! A, I3 Z
    Defined on 1 generator0 r. [& g" _  j# E8 Z
    Relations:) ^1 a# b: N4 y: a  P7 Y- v, @5 c
        2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no ( K' e0 G- c( e% {% t
    inverse]9 u; _* v: V# C
    false
    ; A9 H+ z4 y" T! W* ]* c8 b! @4 z! }7# d) \9 G9 ?0 I6 s4 ^
    147 o/ q) l% f# }, M" u
    28
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    [LV.4]偶尔看看III
    8#
    发表于 2012-1-9 20:44 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-10 11:23 编辑
    9 @1 N( [5 w2 I' q6 a% ~& |" `+ H; P
    + r' O) M# W% S  z; T4 i5 ` 11.JPG 1 ?2 B' X3 C! _( R* Z. U/ s" C. i

    : l# {5 F8 I. M! ^ 3212.JPG
    ! i  f# T; m' t! c& r) ^; h7 w# c8 c3 w( r& \: D; h; h* r6 A$ [
    123.JPG
    0 y, N- t9 E9 }" p9 f0 ^5 T4 N9 |* s+ H. Q
    分圆域：) H& b& h- t" x  C2 c$ b. k; [
    C:=CyclotomicField(5);C;
    # }3 l* P! M7 E0 k8 R8 b4 I3 HCyclotomicPolynomial(5);
    9 Y5 H6 l! |, ]) K/ ^* |C:=CyclotomicField(6);C;
    : u3 E# G$ ~9 Z5 B% G1 I. a$ gCyclotomicPolynomial(6);
    $ M: N7 K7 R% V1 ~CC:=CyclotomicField(7);CC;
    : M$ A( T0 d7 z" XCyclotomicPolynomial(7);2 N/ `+ `! G5 J5 M  W7 y
    MinimalField(CC!7) ;
    1 M: m6 g( a" k$ P, h" O9 tMinimalField(CC!8) ;0 A  ~- q+ d7 b7 V; i: y# o
    MinimalField(CC!9) ;
    : J2 r6 ?+ }( x8 |- C- uMinimalCyclotomicField(CC!7) ;
    / z. _1 D  q- k0 W/ vRootOfUnity(11);RootOfUnity(111);
    ; G$ T' S9 t2 q( sMinimise(CC!123);
    4 W( t, O2 U( K2 uConductor(CC) ;, e8 p9 q& L, Z8 G1 M
    CyclotomicOrder(CC) ;
    ! P- |" [2 t% }- n, n! {% _
    # l! R. o3 [* j4 J  \6 i" }CyclotomicAutomorphismGroup(CC) ;5 b, b3 G+ H+ G! C- E# o

    ; B. |$ ~/ G" \8 Q  u  B$ [Cyclotomic Field of order 5 and degree 4& ~" Q; Z1 R6 W5 _7 |, H4 w
    $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    $ |4 C; l4 x; N# {& n2 `9 F& Y, \Cyclotomic Field of order 6 and degree 2' G* t$ E& K' o" E
    $.1^2 - $.1 + 1
    , r# R; q2 b) p! [Cyclotomic Field of order 7 and degree 6: t' r! v% T7 g; P0 e6 W1 e
    $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1- D( O; c0 S" F
    Rational Field
    : B- o# ]- d; u0 N/ n# KRational Field3 P, d2 E1 i, {; r6 U! N* _
    Rational Field
    5 [( [. ~1 |) I3 f( ERational Field& R8 k+ f1 [3 a; A  Y+ z! ^
    zeta_11
    " ?. K8 h' z4 l9 r/ |- nzeta_111+ |# l1 v) [2 v' f# S  k
    123
    , W8 d2 C; T+ n: L4 ~3 h7
    ) h8 K$ r9 r2 O' ]8 l+ G$ u7( |/ s; B# V) U3 h7 k" G
    Permutation group acting on a set of cardinality 6) P, F; e: r, Q
    Order = 6 = 2 * 3$ }- _! G- P( |$ f* x/ U
        (1, 2)(3, 5)(4, 6)
    5 w1 A1 w. @4 r$ u% s    (1, 3, 6, 2, 5, 4)
      v- F: z! s) IMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of ( u# [+ [' t  `2 X; U3 ~' l
    CC
    8 M6 u( M5 \7 S1 E2 \$ {Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, ; Z% v( b% H8 y9 U
    Degree 6, Order 2 * 3 and
    ' F9 f. R* H( H! \- xMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of : N% H8 c4 l/ m$ a
    CC
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    [LV.4]偶尔看看III
    9#
    发表于 2012-1-10 11:34 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-10 17:41 编辑
    5 K7 s' X* V* a; J0 p1 N
    lilianjie 发表于 2012-1-9 20:44 0 S' k9 a4 O3 j( L
    分圆域：
    8 {* k5 e$ P( `( _1 d+ ]4 ]* b: BC:=CyclotomicField(5);C;9 I$ M# E3 I. X7 D% f
    CyclotomicPolynomial(5);

    & L/ @9 s* [/ d% P% h  J. F
    0 ~" ^" G0 }+ s" r1 e6 b分圆域：+ e2 _: h' R, @, e8 u
    分圆域：123
    5 x4 v) _; Q/ h) {2 p1 W4 a- G* a. B' F. n/ A' e) @1 m/ v! D) @
    R.<x> = Q[]
    . F, y$ E8 M; m6 M2 n  ~/ SF8 = factor(x^8 - 1)
    * l; Z; V  `4 A& V/ I* ?F8
    + s7 n) S" G: H9 w9 P  k
    : ?8 f0 }, L6 }9 ]2 I2 x. J0 ](x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
    : H9 n4 |4 d* V0 u2 F7 s( `$ U5 [$ P: w7 H2 k1 Z9 e
    Q<x> := QuadraticField(8);Q;
    / l/ w8 i; r5 A( J% {- c. MC:=CyclotomicField(8);C;
    ! D3 E6 O3 c+ P3 H, P& K& d* `! k5 ~FF:=CyclotomicPolynomial(8);FF;
    # i. k/ X0 T% R& \) x7 W4 Z7 X+ L
    ) W& k  C, M; V2 b4 U4 K" y  n2 LF := QuadraticField(8);
    * M5 q6 y( |; v0 T8 N. {F;
    / m) {: g# \, z9 @- qD:=Factorization(FF) ;D;
    1 \. W1 N2 \$ UQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    , f$ Q/ T& `) _$ d0 zCyclotomic Field of order 8 and degree 4& q9 ]% O- |4 K4 y& M. ?
    $.1^4 + 1) `& {: F0 z: o
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    ; Y1 j  k2 ^" r+ O) e1 q+ W[; r) E% C. o* q1 F( a6 Z, q
        <$.1^4 + 1, 1>5 I( l5 X9 ~5 F, Q1 L" P3 x# S# c2 C
    ]
    , t3 r1 h! p2 _0 n
    3 E- ]2 N  Y2 x9 ^R.<x> = QQ[]
    9 j  s; ^* C# R* r9 T; L& q+ rF6 = factor(x^6 - 1)+ I/ y# w& I4 S( r6 P: E
    F6
    0 p9 B" ~) a% B
    ' u( m$ {# y5 T(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) ' }* D9 q8 ^9 y4 L

    . a6 k& |, _7 a! V6 s) m$ NQ<x> := QuadraticField(6);Q;* z9 z7 l9 f' F% @, x2 z
    C:=CyclotomicField(6);C;" U4 m6 A: `5 a1 ~
    FF:=CyclotomicPolynomial(6);FF;2 d( U# C( R8 U7 i$ V

    / D+ B4 s1 i1 s: {" R, D2 wF := QuadraticField(6);0 _4 k) p- o9 Q6 `( q$ C* w
    F;
    3 x- U. {5 r- J; F- CD:=Factorization(FF) ;D;! c' ]4 V, Z1 ^9 z6 g
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    2 R3 `3 K. E3 N* ]6 cCyclotomic Field of order 6 and degree 2
    7 {. V5 t% t9 t- X; F$.1^2 - $.1 + 1
      V; g0 t3 X4 OQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field* E% E% `) f) a: F6 g& K; N- `
    [
    7 o8 w3 t! z0 [0 W    <$.1^2 - $.1 + 1, 1>
      {* c2 A, {* N, H]
    ! ^: ~8 J8 b- m# Q3 I
    + n% z' O( v! r1 i0 g+ G! u0 {R.<x> = QQ[]
    " w& W7 W/ `# H) R- kF5 = factor(x^10 - 1)
    1 A8 S6 Y9 F# L/ o/ _$ dF51 ^9 Z. [. h" b- C5 h1 P4 K
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
    8 a8 X; s; q5 S  c0 B5 n1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)6 N- y1 U2 t1 A; d$ {3 Q
    # M" `2 Q- V% U: {" ], Z0 X3 a( B% n
    Q<x> := QuadraticField(10);Q;
    $ e1 J  l3 f! g, b9 ^( eC:=CyclotomicField(10);C;
      Q' |% U! H8 W+ UFF:=CyclotomicPolynomial(10);FF;2 m- x1 I! M6 i( o  u$ _
    7 Y# B2 \( @$ _$ b6 {
    F := QuadraticField(10);
    # Q' \, @4 h3 t; A  ?8 kF;$ |3 S: d& e. p
    D:=Factorization(FF) ;D;5 Z/ t+ R1 A, B6 a  D1 c4 e8 P2 f
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    . b( e2 x' I. OCyclotomic Field of order 10 and degree 4
    5 |5 D' z* Z; ^% s; z! e: `$.1^4 - $.1^3 + $.1^2 - $.1 + 18 A. f! O& Y) a/ b$ D+ F1 Y. R
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    8 P7 Q0 `5 Q0 g! z4 h2 ?[0 x& G, Y8 \& v4 @$ _9 X
        <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    $ g& W* q" |/ j7 m2 E$ C/ N6 O]

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    2012-1-13 11:49

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    [LV.3]偶尔看看II
    10#
    发表于 2012-1-10 11:55 |只看该作者 |招呼Ta 关注Ta
    分圆域：123

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