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虚二次域例两（-5/50）

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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
    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
    3 B, o# ?! ^0 z( O$ I7 P
    6 O3 T0 A) R8 H) uQ5:=QuadraticField(-5) ;/ @2 l( o8 ^9 R3 Q( V
    Q5;
    6 Y: R; `# V; D% p. x# Y( ]* ^2 j+ |. N* H. u7 L( C3 l5 d
    Q<w> :=PolynomialRing(Q5);Q;7 c! q5 t  k" Q3 D4 ]. [
    EquationOrder(Q5);6 e+ S8 E# e" f+ V6 Y5 D% q0 F
    M:=MaximalOrder(Q5) ;
    ! V) S+ w( X" B$ N% eM;( B# q' @+ x% \  R; m8 Q# o
    NumberField(M);
    ' p7 ^7 i8 V/ `0 t- rS1:=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;+ e" @4 X( l  w- W7 x2 L9 b! i
    IsQuadratic(Q5);# l  j6 i0 E9 ]
    IsQuadratic(S1);8 W( @2 @/ @0 m6 ^2 I2 D
    IsQuadratic(S4);; }5 M+ u2 e& b' V) m) t, L
    IsQuadratic(S25);0 e2 R0 Z8 _' N- `. l2 I$ h
    IsQuadratic(S625888888);0 J3 W: h3 P2 W% @
    Factorization(w^2+5);  
    ! N5 w9 k% V- o2 F4 FDiscriminant(Q5) ;
    9 J% m8 E, o4 rFundamentalUnit(Q5) ;! Q# Q1 \4 ^/ i! u6 R# U! Q/ P
    FundamentalUnit(M);) _9 f5 x$ Z$ p; t- _0 z
    Conductor(Q5) ;
    , [+ F- `( M- o6 e; Z; x4 s& |* l& \: o
    Name(M, -5);& s  M: [- e( V1 l' ]1 {- J( j+ C" N
    Conductor(M);' i) D% y; v* S
    ClassGroup(Q5) ;
    0 c! z3 L$ }7 a* H6 DClassGroup(M);
    ; s8 y/ f" c5 jClassNumber(Q5) ;. n0 k- y- Y0 N( f( i; ]; e
    ClassNumber(M) ;
    - Z8 ]' L' S' ePicardGroup(M) ;
    + u; E- I2 z3 z/ u; M/ TPicardNumber(M) ;
    3 U- b1 y0 h  K0 w# u2 F+ D/ f0 b
    ; N" |2 }" _0 O! Q' @2 uQuadraticClassGroupTwoPart(Q5);
    7 U  m" Z" E6 \$ Z: ZQuadraticClassGroupTwoPart(M);" A) `8 r/ ?7 H! ~- b3 d: D- c5 s
    NormEquation(Q5, -5) ;& E+ ~1 w, V6 N! @
    NormEquation(M, -5) ;! d% C+ ]0 ^' c3 \! \) R4 a
    Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    8 s/ [/ E& J" sUnivariate Polynomial Ring in w over Q5
    ) Q% m# v, `0 e4 d! KEquation Order of conductor 1 in Q5
    . y3 N: c' K7 A$ S% {# E1 m/ o% SMaximal Equation Order of Q5
    6 v7 Y7 b1 R( _$ B+ C7 q4 h% rQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    " H7 j2 S+ c  k: l: g9 qOrder of conductor 625888888 in Q5& Z+ h. d0 `3 T3 {1 G" Z4 ?
    true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    $ X9 s4 D9 F  j1 q) I  ?8 Ztrue Maximal Equation Order of Q5
      H$ k' g* c4 T( Q# k" @- ~- |true Order of conductor 1 in Q5, J9 W3 n+ |6 q; `% Z- y
    true Order of conductor 1 in Q5
    8 f0 p0 B. f/ ^2 rtrue Order of conductor 1 in Q5- q4 x9 |8 I; a
    [0 U/ I: j- ~2 t# K- r/ z6 b! N5 d
        <w - Q5.1, 1>,4 e2 ~; I+ e( \3 F' _/ z
        <w + Q5.1, 1>0 I4 S0 ?. w6 F  E
    ]2 Q" r" N! p) H0 x- u: I9 E: K
    -20
    $ h; {# c, H, B+ ^4 j% I* m2 ?+ M% \1 O& r, _. E
    >> FundamentalUnit(Q5) ;, M# H5 W; x3 ]# q: }# k$ c
                      ^. Q0 j* ?: E5 h7 e3 L! i0 S3 {/ I
    Runtime error in 'FundamentalUnit': Field must have positive discriminant" A: l. N8 f+ `- Y) N
    8 o) ^! H, R( R8 }' g/ c
    5 h. P* X: Y+ e% g) a
    >> FundamentalUnit(M);
    1 v' H2 M( l( T; `9 k3 L' k2 }  l1 j                  ^
    5 ^* `4 ^4 C: k; E4 |+ l+ LRuntime error in 'FundamentalUnit': Field must have positive discriminant
    ( j: w. T+ }5 {1 ~! T4 s
    7 e0 D; T8 m: j' j/ G20
    # g1 P2 P( `$ s- e1 h( j: Y- O( C$ t  Q0 z: Q/ ~& V
    >> Name(M, -5);1 I# b. k. h* z4 C% [$ ?" J5 {# j& K
           ^( p/ f  s6 t: P! O$ |# X( i& \
    Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
    / ?' N% K6 M; Q4 d0 g, N
    8 M0 v6 s. F, [- Q. T15 Q  C% m4 C  M
    Abelian Group isomorphic to Z/2+ w1 _' Y7 ]8 @) A& j4 \! l
    Defined on 1 generator6 _' |1 k! D( C, M  L
    Relations:$ e7 ^9 s9 e/ p  x
        2*$.1 = 06 }; o( C9 J) f' c. n
    Mapping from: Abelian Group isomorphic to Z/2! b9 U( M- [) ?3 Q) y, o9 U1 ?
    Defined on 1 generator; X. I( M' ~- I; I' T6 [
    Relations:$ m( k6 v, p" F
        2*$.1 = 0 to Set of ideals of M! x; g. b# h! i2 [2 H' d5 {  x6 n$ _
    Abelian Group isomorphic to Z/2
    9 i, [- F9 _& b8 GDefined on 1 generator
    ( E7 J0 O4 e% V+ o  E1 w6 c8 BRelations:" Y3 ^3 U9 D$ Y9 |* `7 d
        2*$.1 = 05 T, N; j; |2 [
    Mapping from: Abelian Group isomorphic to Z/28 N/ e2 N# A# o" o# M/ w5 G  e
    Defined on 1 generator
    - x0 Q# B9 {. z2 @Relations:' ~+ E0 @$ @4 A& _0 u# U* q
        2*$.1 = 0 to Set of ideals of M
    4 @+ E( a4 _* i* H% d: v2
    / x6 i7 F! G2 h! M# {) _26 B9 t% i9 N+ {
    Abelian Group isomorphic to Z/28 ]( q2 P7 J$ A6 w3 S% v
    Defined on 1 generator
    5 D5 e- m7 ]0 X3 O! H1 zRelations:. l$ v( Y" X/ @! J8 o( }
        2*$.1 = 0- A: @# {1 A1 Z7 i8 A% u0 V2 x
    Mapping from: Abelian Group isomorphic to Z/2
    2 L8 N) t7 U1 l* F3 x5 m- j" W+ Q2 sDefined on 1 generator
    $ p% U+ B) L( ?+ ARelations:& ~% u2 a3 E3 O& Y; X
        2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
    : |1 e+ V& {& i1 n3 N# g2# w, `1 q* k' l0 R8 l  x
    Abelian Group isomorphic to Z/22 C. q- ~# X/ P0 w( j* p9 z# R
    Defined on 1 generator
    1 m8 ]; r- N* V! z: |% XRelations:
    7 ~: i" }$ b4 I    2*$.1 = 0$ D/ }: ]# x% W: q- X$ K) A
    Mapping from: Abelian Group isomorphic to Z/20 n8 w% l1 X! N4 _1 M
    Defined on 1 generator
    & n4 L- d# |6 ?9 w/ p' wRelations:
    # M; c1 a( _8 B( z# r    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no : }) t% L8 c( `! u) j
    inverse]
    $ j, k' i4 A' P0 E! D4 |- F. z$ n1 `, OAbelian Group isomorphic to Z/2
    1 Q  }$ ?- Y% v7 EDefined on 1 generator
    1 e: f2 B' i  K2 ORelations:$ G6 K$ k" o0 e% K9 ]' h; e5 b
        2*$.1 = 0$ j4 l' E+ B6 ]
    Mapping from: Abelian Group isomorphic to Z/2
    % }4 S6 D- h: PDefined on 1 generator( I3 U7 ]6 d  {' |7 w6 ~
    Relations:
    ' n) ~4 _% Y' f5 m( E- y2 |    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no ; U7 S" ]# r, }6 q) N1 u5 u$ c
    inverse]# S- }/ ~, H) a+ l3 d
    false
    ( X3 T% Z# V" j1 A+ ^& pfalse) G2 d8 V5 Q" T7 i. g8 t
    ==============
    + N4 \+ z7 ^/ k) y; T0 }
    . e" V$ e/ p) V
    $ k* e( {0 x/ [% S4 l1 W/ F2 W5 ]& W! ?Q5:=QuadraticField(-50) ;2 X, V2 \  C7 _5 y$ ^
    Q5;
    + w0 o! L5 d! N. [, I# P
    / W0 n2 e/ ?  A2 IQ<w> :=PolynomialRing(Q5);Q;
    ' p; k. i; k: O: m7 v+ |EquationOrder(Q5);
    : V) u, K$ k; W5 aM:=MaximalOrder(Q5) ;
    ( K1 n( Y4 ?0 c. l1 JM;7 r6 ?3 u6 T: L: `# }
    NumberField(M);+ ~* d4 y. d+ H7 k/ `( ?3 L$ Y% z5 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;5 Y1 `/ |% d4 M# _( g2 u- X' T9 T; o
    IsQuadratic(Q5);
    7 s0 k1 c2 t' @0 p1 oIsQuadratic(S1);/ B+ y3 D" c, S; E4 M
    IsQuadratic(S4);
    & Q2 ~* B0 P7 ^2 i- YIsQuadratic(S25);
    0 b* B1 a9 i# P7 cIsQuadratic(S625888888);1 p# }  l, F7 c6 [1 R9 d
    Factorization(w^2+50);  
    / M$ _5 j- F3 {6 t* VDiscriminant(Q5) ;
    5 ]: ?* o# @# b; W3 D: _. i* jFundamentalUnit(Q5) ;
    : O# a3 P* m+ g+ ?# I( ?: _FundamentalUnit(M);
    % C% A# ?1 \; _% D9 {/ ^Conductor(Q5) ;8 @# I5 s& Q$ j$ M. a6 v
    8 G) h9 X" u- M) ?
    Name(M, -50);
    ( {" m- L6 e4 T) v! M0 w. nConductor(M);6 c5 W% c( [! v! ^) D; B
    ClassGroup(Q5) ; ! ^9 ?% Z" p: i0 s# L
    ClassGroup(M);4 a! ?2 y" j8 J8 g! I
    ClassNumber(Q5) ;/ q$ s; y! |9 @- w" C
    ClassNumber(M) ;
    5 ?" s. |* Y7 E+ ?7 gPicardGroup(M) ;
    ' ]. c7 J- v0 h' m" z4 M5 \PicardNumber(M) ;# e& V1 O- v- D/ l1 a
    % V; `' n; G7 s6 j/ W' C
    QuadraticClassGroupTwoPart(Q5);4 F) ]+ y/ e! x2 G
    QuadraticClassGroupTwoPart(M);; p" A6 r4 A7 ?
    NormEquation(Q5, -50) ;. P/ j1 e% V% W% s- E
    NormEquation(M, -50) ;
    % V& ]  ?0 o% X7 D
    4 [+ W: v# F" X. tQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    ' }+ D& J: q" Q$ Z% UUnivariate Polynomial Ring in w over Q5# Q( J+ M/ Y4 Z2 ?- _6 n; M. y
    Equation Order of conductor 1 in Q5% ?! g" ^5 Z+ f6 o; W! D6 T' ~  h
    Maximal Equation Order of Q5" V+ f7 n8 _$ n7 d2 S+ i3 _7 b
    Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    0 `" \5 P0 x  y- s3 u1 ]+ BOrder of conductor 625888888 in Q5
    ' u# ~- g1 |( o  u2 R8 ~true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field( s' |* @3 N+ x2 w
    true Maximal Equation Order of Q5
    9 L" e# i$ d' v, ]; Z7 T/ btrue Order of conductor 1 in Q5. i3 T+ D2 M; W9 v' }
    true Order of conductor 1 in Q55 K1 g! g4 @6 k
    true Order of conductor 1 in Q5
    9 {. b& t( D: W$ }[
    / h+ U" k9 o2 J' x0 i( \; L& K  u    <w - 5*Q5.1, 1>,9 h+ U7 z) `2 m8 h0 a" v$ m
        <w + 5*Q5.1, 1>
    9 U+ U& R% v1 T- v% V]3 }5 L! q9 p  f+ e$ L( d. F
    -82 U) @. Z' B5 L7 c! ]8 g6 h; M

    : a- x" n5 t* |>> FundamentalUnit(Q5) ;6 I, o  ^4 r( T7 G/ E; V
                      ^5 a  o$ `2 O2 Q
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    9 w9 \( {; b& z8 @/ x- M
    2 v5 u4 i8 V& Z$ X0 N
    , S7 \7 M( [& d+ V4 S* O) r% s% C1 r>> FundamentalUnit(M);
    ' A+ l% {* \2 j( z% \                  ^
    7 Z( K" s$ x+ t; K! \, G- p+ VRuntime error in 'FundamentalUnit': Field must have positive discriminant" e( X5 K! s1 i" Q9 j  A& F. G
      W5 M) e" Y+ u8 S7 d* n- s6 f" u
    8
    2 c4 L2 s. S- X" n; {8 ]0 }
    6 @) X, i; L# \5 C1 g! y1 @>> Name(M, -50);; s9 t0 K% y4 i. F
           ^  k: J8 m" H6 [- u9 K9 y
    Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
    7 L3 R2 \$ p( j$ N- V1 O& g/ O5 r$ J/ m- N
    1
    4 u' Z# U' z5 V8 BAbelian Group of order 1$ p& k) M+ e  D! L+ j4 S% E
    Mapping from: Abelian Group of order 1 to Set of ideals of M3 E1 O& P0 V- S$ H1 o2 ^( \; C
    Abelian Group of order 1
    " H+ \) c% J: t6 Q6 h8 [0 b( L/ QMapping from: Abelian Group of order 1 to Set of ideals of M
    . @* o% b3 c1 }& b9 y, r1$ e0 v2 w2 F5 w. p. f
    1
      m3 A$ k% z" k1 m4 [Abelian Group of order 1
    " a% ^. D, t) Q1 a$ ZMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    3 g, r: L) o; t! `  N) j, c# binverse]
    * \7 P. g$ h  Y  F: r1
    / H3 L3 U6 l/ ?  ~+ x" y: ~5 m! YAbelian Group of order 1
    5 }7 C: _3 ]/ ~2 [6 n+ w# gMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant; ^( U. F; b. V
    -8 given by a rule [no inverse]# X  Z. Y* I/ {5 U5 v
    Abelian Group of order 1
    4 c& D' n, U; t4 E$ R; T& Z% p0 f, f/ @Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    2 _+ q1 H' M, i+ C3 d1 K-8 given by a rule [no inverse]
    , t: n# m0 g. F% c. ifalse8 @2 q! C9 ~/ ?- ~0 `
    false; h6 d) ]3 H: w" U9 U$ p* Z( V
    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：* |. J0 F5 n, s

    - e, W% V  B  R; W) g1 mQ5:=QuadraticField(-1) ;) p: r1 P$ `8 ?( J% R
    Q5;
    7 ^5 A, D0 i! w6 @0 }2 I& q1 t, w3 G
    Q<w> :=PolynomialRing(Q5);Q;
      }  g2 o" O! w+ C) I8 m+ `EquationOrder(Q5);9 S, _1 {" s/ q; @
    M:=MaximalOrder(Q5) ;
    2 b" e0 g3 Z  X# i7 w; J# k" wM;
    # l' k  K, P" R  _7 R! INumberField(M);
    + X1 G: d8 S# S& O6 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;& ?  _& O, ^- R1 S7 c1 e
    IsQuadratic(Q5);
    4 ?5 G: M% I& e, n6 ~IsQuadratic(S1);8 P* k0 Y" S4 L( _+ D7 f: [) z
    IsQuadratic(S4);
    7 c; ?9 K' x5 [7 y0 kIsQuadratic(S25);
    3 @, r6 X& u, ~( ]- C6 B/ w& {IsQuadratic(S625888888);: u: k/ H6 u$ k  M+ [
    Factorization(w^2+1);  7 b( W* ~) u- @  M# R
    Discriminant(Q5) ;2 ?* x$ r7 _' q' q, |- R! X, E, o
    FundamentalUnit(Q5) ;/ L5 t: w5 Z9 R8 i! ~
    FundamentalUnit(M);. d# y- e9 b, r& B4 Z8 A
    Conductor(Q5) ;
    $ x) S' z1 @$ w& i/ q* y2 r5 z+ E
    Name(M, -1);5 B, T- L! E. w' P+ T# l
    Conductor(M);7 U7 {* a, \7 G: s6 I- Q
    ClassGroup(Q5) ; ( u- q4 G* K0 t# |  k
    ClassGroup(M);$ }6 c, \1 G% q( @. |" r3 e
    ClassNumber(Q5) ;
    * d2 }+ h6 D7 \' e8 _& }3 YClassNumber(M) ;
    ' |$ R3 G) y) d. RPicardGroup(M) ;
    2 \( @7 c0 g' y' O) j9 }PicardNumber(M) ;( x: c4 I& d6 G- e

    0 R! N$ z8 C) V& \9 x/ wQuadraticClassGroupTwoPart(Q5);
    ' ~8 j" l/ T# U3 \5 z2 IQuadraticClassGroupTwoPart(M);
    2 w9 R% m1 v- N1 n+ L0 j' _6 Y, GNormEquation(Q5, -1) ;
    ! d) u) A. ]. Q) ?8 O  ]! S. h+ uNormEquation(M, -1) ;
    ) h' @1 |% `. O3 D- c5 `  p) Y8 k! L/ u+ K, E/ c% ~% }; @
    Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field0 p& A- C) A) I
    Univariate Polynomial Ring in w over Q52 Q7 n' s  S$ o) a
    Equation Order of conductor 1 in Q54 t- {! f; B  L" J3 a- C5 G
    Maximal Equation Order of Q5
    7 P9 X* ]/ r! _7 T2 l/ B2 p9 FQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field/ y- `! k% S% D4 }) q: R0 g0 I
    Order of conductor 625888888 in Q5
    & O( Y+ ]; O, M* h+ F5 dtrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    ' r2 _+ z4 o2 l6 L2 G. }7 ptrue Maximal Equation Order of Q5, v6 W  X9 O" n. h. b
    true Order of conductor 1 in Q5
    5 o$ Z7 M9 V! v8 F8 Y8 Ftrue Order of conductor 1 in Q5( U7 G. z7 U1 F" H
    true Order of conductor 1 in Q5
    7 A# Y9 R0 l' W% ]0 h6 F. S[1 q$ X: Q! }) `) V' T
        <w - Q5.1, 1>,. w8 C$ d. I" o6 K% I0 k/ l0 r8 N
        <w + Q5.1, 1>
    , d9 x: x4 a9 ?]
    ! O6 Z; o7 K# W& B4 E  U, W* m2 X8 w-4/ M$ P$ G1 b1 d, w
      U, }1 Z! A' l# @# O; x* ~9 |
    >> FundamentalUnit(Q5) ;
    5 X3 F- G& L2 q6 Y                  ^
    5 }" o" J. ~$ r6 P+ |% sRuntime error in 'FundamentalUnit': Field must have positive discriminant
    " X# s& f' c/ r% _8 ~: _* [, Q- k( s3 a
    4 T, j$ f6 {/ C( [# }# c
    >> FundamentalUnit(M);
    # X  W  E. U. A% F# K                  ^$ k" H- ^2 {# M( l% S3 x# s
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    6 @4 O& a( k6 s8 H9 ~8 l/ {; |# J9 k* L, l
    4; L# L( f/ l: z, d5 |  o

    $ X! H% u, f2 y1 ?' r% |>> Name(M, -1);
    % w- A; M' U2 Z7 b6 v# E       ^* t2 H; y3 M- w+ ^' j* x
    Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
    : q* S( Z( b2 r( `9 |: \  }$ ~% G7 G  b; D. r9 u% G
    1& F& |8 P% [, A5 E4 d
    Abelian Group of order 1
    , W8 i* `: d# O& iMapping from: Abelian Group of order 1 to Set of ideals of M) G. a7 c, E7 r( K9 k. S
    Abelian Group of order 1
    6 L6 S" X/ l' E# qMapping from: Abelian Group of order 1 to Set of ideals of M( f. ~; S# F( \; s3 ]+ A4 a
    1! y! T. D8 f' g( r6 E
    1+ |$ |: D9 ?9 A  [1 i  f) |5 B- b$ _
    Abelian Group of order 1
    " V) D2 i( i9 J- O& wMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    ; y! F6 Q$ u2 k) E* n# Pinverse], H2 J7 T/ A: n9 Z, t# W2 u
    1
    9 N- t& ~' m+ X; d; ~- xAbelian Group of order 1
    * I, x) z+ h: v7 i: |  hMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant* P" G/ ?) q6 S; T/ d5 ?
    -4 given by a rule [no inverse]( W+ ?& {$ P9 P' f: Q
    Abelian Group of order 18 V3 Q$ [. B4 J2 ], s( @
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant' S; c7 j$ @3 ]. l
    -4 given by a rule [no inverse]
    # O! V/ P' Z: d/ S1 Pfalse
    . ?; ?4 ?: V9 nfalse
    ) [8 @: j. v0 [3 Z! ]( @. T===============
    1 R0 r; a0 Y5 D% }
    % G6 Y/ ~# m: V0 Z" DQ5:=QuadraticField(-3) ;
    9 u* b" M' i2 l' f) iQ5;7 Q9 x0 @1 ^+ Y: z6 i( S  {2 Q# t
    " v6 |' g* g$ N: \+ P% ~
    Q<w> :=PolynomialRing(Q5);Q;
    9 |. {0 {/ H' e- s5 L3 E5 PEquationOrder(Q5);: A0 K; e, I$ Y& T8 a0 L5 J  q
    M:=MaximalOrder(Q5) ;- N: D0 t# X2 l: a- M( t; Z! S
    M;
    ( Z2 a3 o" N3 K' vNumberField(M);
    + s: s0 v# t  i1 US1:=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;
    - m0 L# m0 l2 x  S& K0 s+ pIsQuadratic(Q5);
    , K/ D1 X1 w) O5 ~# y! qIsQuadratic(S1);8 B4 |  }, Q( G' f% Z
    IsQuadratic(S4);' o. O6 \  |3 Q' v
    IsQuadratic(S25);! t3 D  ?( S+ ?+ y& |
    IsQuadratic(S625888888);; X; C9 p2 I& @% k0 d% u+ ^, C) n
    Factorization(w^2+3);  : w4 x, w/ L  s6 n7 i8 S- r
    Discriminant(Q5) ;
    : J0 O2 W" v" k; ^2 j5 uFundamentalUnit(Q5) ;
    9 s# \. Y( H0 W& W  C0 FFundamentalUnit(M);  N4 E( Y8 c0 v/ k
    Conductor(Q5) ;
    " f- e8 t' Y  D5 ~& q* h) x! b5 ?8 h1 w; f& x0 r$ Q
    Name(M, -3);
    6 n: r+ r7 Y. t" m( cConductor(M);
    $ h: F8 X$ S- n, r- D  ]ClassGroup(Q5) ;
    $ b, x6 K7 O+ hClassGroup(M);
      P( q9 \1 e& [1 T( p( T# q3 y8 @% jClassNumber(Q5) ;) I3 K2 O( q, A& e, T7 Y' [
    ClassNumber(M) ;4 \1 X, ]6 x5 a# v4 K9 g0 C
    PicardGroup(M) ;/ y5 q, }/ \4 ]( J
    PicardNumber(M) ;7 z* V, Z# O( r1 M: O5 u* {3 Q
    2 @8 I  X/ }, C# e1 r4 O6 H+ g
    QuadraticClassGroupTwoPart(Q5);
    1 [4 s2 W3 S5 b1 c! R2 ]3 {% VQuadraticClassGroupTwoPart(M);6 |/ f1 D6 [; f1 j
    NormEquation(Q5, -3) ;6 v1 }2 C0 c  z
    NormEquation(M, -3) ;
    4 `$ t* y4 ?) l0 B4 a1 M6 v7 p
    + o/ r1 o& w+ C  K9 [; _& aQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    1 `& C& V! ]1 C+ s/ NUnivariate Polynomial Ring in w over Q5
    ; @. M0 f6 b5 N1 r- V1 B4 B1 `Equation Order of conductor 2 in Q5
    + u* U+ u) z2 p8 R6 GMaximal Order of Q5
    - B# G. o7 ?0 B2 F4 k8 p# U# S$ \6 HQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    & h5 ~: i9 T) V2 w/ _/ W2 a- BOrder of conductor 625888888 in Q5( d6 K' k: F1 j6 D
    true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field! ~; P% J# v/ \1 N3 F
    true Maximal Order of Q5
    9 Z% I3 T3 I, d4 {2 z% s, t3 rtrue Order of conductor 16 in Q5+ \: R6 A0 U* R$ N$ {
    true Order of conductor 625 in Q5
    $ x' ~, v6 L* P5 V& atrue Order of conductor 391736900121876544 in Q5
    ! J% S/ v7 I- f$ f0 u) \( Y! h: ?2 v[
    ( V9 E4 m: {* E; v$ [, v( k1 V    <w - Q5.1, 1>,
    : ]8 u& F' k; `& v5 l/ {    <w + Q5.1, 1>5 ^0 S" y: J0 D* z: n  f
    ]
    . w, @4 }# f* p2 i6 Y-3
    ; v6 \! H2 y  ]1 F; d+ u5 y' ?2 C/ e" f$ U+ k
    >> FundamentalUnit(Q5) ;0 `/ ~0 o% b0 j* @
                      ^+ L2 {8 K$ u9 T* \) H
    Runtime error in 'FundamentalUnit': Field must have positive discriminant; G* o! \, `, h6 B  g9 ^
    6 g3 ?6 `  h. {2 T) d/ w

    6 c3 q' Q: C: _2 e>> FundamentalUnit(M);4 V: x& b% Z5 B0 s/ _1 d
                      ^
    8 u0 [4 p/ M  BRuntime error in 'FundamentalUnit': Field must have positive discriminant
    $ V9 ?6 ?0 i* p7 v4 D
    + z" ]! d6 v! n) o1 F* b/ d. {! S3' W4 N% T9 U% W1 B
    . w  |7 d/ A) f5 e0 V
    >> Name(M, -3);
    2 i& X* e8 B9 d6 S& G       ^2 Y3 w4 ~$ `) @* r
    Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
    5 o5 ~: I/ V; A- O4 C2 m5 Q( l2 l2 U- i
    1% O5 |5 p$ ]4 D7 X8 B$ L
    Abelian Group of order 1
    , T" u) ]- `$ NMapping from: Abelian Group of order 1 to Set of ideals of M
    6 ?2 s1 X* p' f* FAbelian Group of order 1; e+ J: d  W% _3 f7 J9 T
    Mapping from: Abelian Group of order 1 to Set of ideals of M/ R$ q9 [/ Y0 W+ X$ R& Q- u) l/ h  {
    1, S0 G; z5 b1 {9 y
    1
    2 F; z' y8 M! q& [! |/ m8 O" nAbelian Group of order 1& y/ m& `9 l8 T9 i9 C
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no# l3 R: M9 x2 W9 j7 ~+ `
    inverse]9 H9 c! I+ v) R4 {1 z1 E" \0 |
    1
    8 y- l' p! `! q4 FAbelian Group of order 10 K9 _8 n7 P. y3 \# n
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    4 F. K3 i! O1 z5 B! S5 t-3 given by a rule [no inverse]
    $ j0 ]" E0 d+ Q: i6 O1 X" RAbelian Group of order 1  v! d' E  N0 o) a  c$ I
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant; J+ `* N# L# C3 I( a
    -3 given by a rule [no inverse]% p8 \3 |& o0 Y
    false2 ~. L* A/ `$ P. i
    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 2 i3 b& q4 c7 a# E3 Q. v

    3 w# ~! F1 [) ?* c) U* EDirichlet character
    2 q, M' n1 |4 l6 R$ m0 i0 q; WDirichlet class number formula4 x- y: h( n$ |
    & b7 D, G8 K6 q5 p
    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7 x. A5 v5 E8 i: O% E6 m( L

    ) o, h/ m; K$ r! x. z7 ~-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）]=1
    # \; g6 i9 {5 M. c( `7 U7 h/ i: z' j6 f  g1 v% W$ q& [
    -3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
    . B( O7 O9 V+ A: |4 oh=-6/（2*3）*Σ[1*1+（2*（-1）]=1
    $ Y, o' @. v# k7 Q. n1 _3 Y: ?% N7 W) }. K. C1 V. {
    -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，
    8 ]. f3 B4 r! x. U" |! M5 Z9 w( P
    " {4 u) ^& U& k! r- {- U% y1 h3 T+ I, n) p! E& O0 V8 f
    5 z7 E/ p+ t# W( g5 S8 f
    h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=26 X4 R( B! |; w: K. p4 B
    6 q4 p0 G3 [$ W9 A; ^
    ( h/ C  A; c1 q! ?4 |2 Z
    ( ~8 K# \2 W$ b5 g1 g
    -50时  个单位根                          N=200/ f9 D; v) S$ g7 B+ B& J& R
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    2012-1-13 11:49

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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: 241)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 245)

    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: 245)

    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
    + J. _# J, A% ^2 ?) @- l! i, K2 `, h
    . a8 U' f: n) \& W$ \% h/ WF := QuadraticField(NextPrime(5));0 t  _  {6 j5 @8 l( {

    & u5 r- `9 F# V/ \$ _6 xKK := QuadraticField(7);KK;" Z0 O3 m* e  e7 M) Z5 J
    K:=MaximalOrder(KK);
    * k5 D4 _# ~+ C5 \6 J3 ?Conductor(KK);. S$ D0 a. D0 u7 o6 i
    ClassGroup(KK) ;
    ( E% z* A* u8 P3 `9 XQuadraticClassGroupTwoPart(KK) ;+ A* P! \3 s0 {
    NormEquation(F, 7);
    $ G8 v6 c2 r; p2 WA:=K!7;A;
    4 B5 J- H  k# k! zB:=K!14;B;
    " v$ r3 f1 a! R* d0 Q, hDiscriminant(KK)
    # L4 B& @5 W0 z2 i: }
    % a* s1 {4 b9 a. L$ K' kQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field$ @8 `  P4 ~3 [2 z
    28
      Y. X* Y6 c1 G: [Abelian Group of order 15 M3 h2 @, x& X
    Mapping from: Abelian Group of order 1 to Set of ideals of K2 E7 s# d& F3 g8 }) P
    Abelian Group isomorphic to Z/21 }* T% R; `# E' D" ^& H+ @
    Defined on 1 generator5 f& i1 ?( c, _
    Relations:
    7 B; Q* j" W( s+ [* n    2*$.1 = 0  @' a' o5 G5 q7 [$ M( Z$ ?
    Mapping from: Abelian Group isomorphic to Z/2/ `9 V, v5 t7 b) x% o1 V+ t( {
    Defined on 1 generator3 Q2 z. s1 k  F1 b# A3 n7 z
    Relations:, |6 D$ h) [5 H0 P2 @
        2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
    4 f: C, |8 ?: ainverse]0 a- E" Q2 ?3 h7 E' p+ H
    false
    2 n5 U+ o) @+ ~) w* D" [5 c7
    / u  U7 c. h8 k3 \14
    ! g9 g$ F! l$ }) p0 Z* `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
    7 y7 K9 g' J1 W+ A0 j6 k- U- X( T' F3 \& a1 L2 G* Z4 V0 C
    11.JPG
    8 F8 C$ E6 t9 j2 {) l3 R4 e% ^# I! q/ i# Y% {& X( ~; R$ ^
    3212.JPG $ _! G2 E! c" h' u! ?

    2 |: V( o  t9 U. Q7 Z% g( ~7 w# u 123.JPG 1 h8 {5 j' @$ p: k( T, B; `

    + @. |6 c) _7 p8 b& Q& M分圆域：
    " N8 d0 I( d1 f5 Q: @/ hC:=CyclotomicField(5);C;
    7 N' t2 N1 Z0 qCyclotomicPolynomial(5);- D! c2 I# i3 @) e( o! O4 P) F
    C:=CyclotomicField(6);C;! i6 J9 Z/ w( K9 ]7 j$ L9 v
    CyclotomicPolynomial(6);" l& [; o3 h5 ]
    CC:=CyclotomicField(7);CC;
    8 M# g) D7 a# R( I  z) bCyclotomicPolynomial(7);1 C. h2 Z' b1 Q# ~
    MinimalField(CC!7) ;  v- A7 S! {+ D
    MinimalField(CC!8) ;
    ( l0 w) Q- h( |! Q* Z) QMinimalField(CC!9) ;
    / A) d, Y: g5 N5 fMinimalCyclotomicField(CC!7) ;+ K- A: C( D) z4 o
    RootOfUnity(11);RootOfUnity(111);* ]/ v" I2 v$ J: K2 q0 @% N
    Minimise(CC!123);
    ( _' A2 i0 M, m9 yConductor(CC) ;. g9 w4 c9 n! H
    CyclotomicOrder(CC) ;
    + M8 R4 {8 {( [- H
    : U; Z. u; C- M+ x2 B+ KCyclotomicAutomorphismGroup(CC) ;
    6 ]6 B9 ]3 p2 g
    . v3 Z$ f8 F; D- S) x2 }Cyclotomic Field of order 5 and degree 4
    / U: h8 b. R6 n  J# r+ J+ o2 H6 l$.1^4 + $.1^3 + $.1^2 + $.1 + 18 W: l/ s4 C! U6 E, Q9 \. j
    Cyclotomic Field of order 6 and degree 2
    2 y* B' I1 {4 g2 i+ W; E# ?. ^$.1^2 - $.1 + 1* V6 a- p* J# p0 s# Y% z1 k/ F
    Cyclotomic Field of order 7 and degree 65 j3 l* C2 y5 ?. z) |4 B' N
    $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 11 v* F4 y. s# H
    Rational Field
    : I9 U. `. y( m9 j  j3 `7 {" H6 N4 |Rational Field
    0 o* m* Z$ N. ~- }& \Rational Field( h. \, w. k' I8 I7 F
    Rational Field
    3 u% m& ^& I% q2 z, b# z6 P2 i: Lzeta_11
    8 Z$ M2 I5 x% D9 _# Zzeta_111/ x' j% s$ I6 [) m# t! x. `( V
    123
    8 X: {- m' D; k1 w2 F7
    ; t7 r2 g+ U1 d3 H. N! ]' W9 V- U7
    ; b- ?. Y! n. z* K: j2 `Permutation group acting on a set of cardinality 6
    4 \- Q/ w: I. }0 c; T5 [Order = 6 = 2 * 38 J$ q  S( y0 N: h4 F) t
        (1, 2)(3, 5)(4, 6): z( \) Y9 ^2 U
        (1, 3, 6, 2, 5, 4)& {+ u9 x+ I* b0 }' c- o9 {
    Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    ' H# c4 ]) [. uCC
    ) l- z5 I$ ~) H# S' `: fComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
    / f2 W8 ]: M' K5 v. \9 L# ]5 lDegree 6, Order 2 * 3 and1 y$ U  x& n2 R% t5 l7 q
    Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of % e1 y9 k' w$ a' }% H: Q4 u' Y9 i) r
    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 & O- F: W4 g& c5 b1 A& V
    lilianjie 发表于 2012-1-9 20:44 $ x# m0 [0 x7 @1 R4 A& }
    分圆域：  O; V. I) o7 H, @" s
    C:=CyclotomicField(5);C;+ _* i9 z& F& V2 B; L8 ?
    CyclotomicPolynomial(5);

      D5 o$ n6 B3 R, v: l
    . P( Q8 Z3 U) B* q. [3 t分圆域：- m) j# r; q4 z
    分圆域：123  P( r- q9 z7 a' M6 c% E

    6 Z/ M  Q2 ?9 o- J, T: ?& PR.<x> = Q[]
    ' x' ^1 I  b2 @, {F8 = factor(x^8 - 1)
      q1 a7 Y- m" r- e6 nF8- n. g% [, e! N
    ; U5 q* @+ G; K1 S: I4 i. A; l
    (x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) ( ]% z9 _5 t+ M  Y
    7 P& s0 P: t, f1 S0 K" ?+ E
    Q<x> := QuadraticField(8);Q;  Y( H& m! Z/ j: Y+ \, q
    C:=CyclotomicField(8);C;# L% }9 |" [- u6 I  j# _
    FF:=CyclotomicPolynomial(8);FF;
    $ e. q6 w6 s0 ?: e$ m7 q0 R* v5 d: A6 |: }) W
    F := QuadraticField(8);2 u! Y" Z3 R7 S) A+ f
    F;
    / ?' V$ {* a' v& \D:=Factorization(FF) ;D;
    3 i7 o6 B( @8 X/ p4 x5 ^3 gQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field) U; o" _, |& A( J: Q' u
    Cyclotomic Field of order 8 and degree 41 K! Y# ]2 y5 W" ?8 l  T
    $.1^4 + 1
    7 z8 [% a8 O- \/ m' `. zQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    % o/ k, j" u/ ?* U* F[# v, Q6 S0 r  r
        <$.1^4 + 1, 1>
    6 a/ s+ E) f6 q" G8 R]
    1 M- X% b# a8 s! j4 X! D, `; W8 i
    R.<x> = QQ[]
    . z+ `" P- ^) c) ]F6 = factor(x^6 - 1)  h0 _# |+ ]! l) k2 d1 [' L% m. s( b
    F6
    1 n2 C3 S6 U6 W$ j/ e$ `0 z
    % P+ Q% v3 `- A: f; R8 p* d" ^2 c(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
    / }3 D0 t1 n8 w- M! T; d$ M6 l$ Y
    ' g7 z" P# c& c1 V$ M& C0 I6 R: e/ xQ<x> := QuadraticField(6);Q;. ]1 f$ w8 _! p& H. A
    C:=CyclotomicField(6);C;! H! m- m; Z: z/ J- R: K0 k
    FF:=CyclotomicPolynomial(6);FF;" H7 ?0 n5 r5 F( z) N

    ; \$ Z0 s3 I! t# n5 ]- O' }F := QuadraticField(6);
    ( O9 g$ M5 Q5 r+ H' ^6 X  jF;) O& l1 ^4 l& \5 D/ F" e3 f. l
    D:=Factorization(FF) ;D;; ]! N8 i- ^; T
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    # Y! z$ X" _5 {) D( c. oCyclotomic Field of order 6 and degree 2
    ! m  E9 H) U7 W8 D$.1^2 - $.1 + 14 v/ J, _( r0 \+ Z
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
      F# A. l7 s  P7 c[
    4 d" U5 m6 J" [* H    <$.1^2 - $.1 + 1, 1>
    / A5 \' G7 n" |]
    ! I! S7 h7 u( e% N/ S+ s% q, P' x/ `* {
    R.<x> = QQ[]2 o# n) w1 U/ l$ }) y4 ~
    F5 = factor(x^10 - 1)
    3 f4 G+ C! ?& a. KF5
    7 z4 ^# Z+ `% ~" Z(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +0 y+ N0 e+ o2 I- K; a8 \
    1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)) N$ V0 e0 l5 J5 U' v* E& y

    ; t$ {/ N4 R. YQ<x> := QuadraticField(10);Q;" h/ a" `3 ^, w/ H  S2 ?$ l, u; Y
    C:=CyclotomicField(10);C;
    / z. Y( |! \2 P8 I- b* v: Z' \( gFF:=CyclotomicPolynomial(10);FF;
    / \4 q" D# H& F) ^# U; a3 q* g/ P" @( n. ?; L  z2 X- P* E
    F := QuadraticField(10);. n0 t% z: c( A1 x* k7 @9 m
    F;
    ' x" M  y0 V1 }D:=Factorization(FF) ;D;
    0 Y" X+ c; L" sQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    / Z/ r! N" j# b6 RCyclotomic Field of order 10 and degree 4
    8 P7 G' ]4 a1 K$.1^4 - $.1^3 + $.1^2 - $.1 + 1
    ' C3 E6 @/ x0 zQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field* k/ c! J+ V; Z; C
    [
    6 D+ ~/ R# J1 |+ T: w/ s6 K    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>4 z4 m8 s( R) \$ ]7 _
    ]

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    lilianjie1        

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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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