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虚二次域例两（-5/50）

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lilianjie        

43

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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
    : V$ ^$ E0 z4 {) a; h
      u( q9 c: K5 IQ5:=QuadraticField(-5) ;
    9 n) m" C* k. mQ5;) r# s; u$ h& }; e5 j

    & p& Z- w. P8 B7 J8 pQ<w> :=PolynomialRing(Q5);Q;# O: J/ P  M$ a
    EquationOrder(Q5);/ [0 ^. b9 k0 t! l/ c4 u) S! ?
    M:=MaximalOrder(Q5) ;
    4 w4 Q! r9 j  ]2 FM;2 F; |- }1 A1 Z  O* U" }
    NumberField(M);
    ! g& H: L  {' ^: D4 PS1:=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, p9 H  r5 B& CIsQuadratic(Q5);/ F5 z9 d- L" ?+ ^
    IsQuadratic(S1);
    9 Z2 B& u% q# p, m" K: Y5 q. dIsQuadratic(S4);, T1 ~6 N, ]$ ?: G  v
    IsQuadratic(S25);/ c* ]( J  a& x( r; P6 k8 T: Q% ^1 Q
    IsQuadratic(S625888888);
    . z9 Z! a9 M+ e7 WFactorization(w^2+5);  
    & I1 y! A7 z4 v! r5 K6 g! BDiscriminant(Q5) ;
    4 Z* I- a, b0 OFundamentalUnit(Q5) ;' j, \& D+ O4 Z) p
    FundamentalUnit(M);/ Q* _' x( k# W2 U/ @. W
    Conductor(Q5) ;
    . o$ J2 q# m  e" R
    1 S' V: J/ |6 t# xName(M, -5);0 J; ~8 K* A% H1 k1 E
    Conductor(M);
    0 \4 `# y$ c# R' x6 `( F. `8 UClassGroup(Q5) ; ! Z- Z2 O: y+ J, ?( _+ R- Y& [" o
    ClassGroup(M);4 k5 a) T, C/ E; X, G
    ClassNumber(Q5) ;
    " y4 K$ H% h6 ?ClassNumber(M) ;. j/ p, c% g' ?5 N3 E
    PicardGroup(M) ;
    7 n8 _8 J" h4 O* @; aPicardNumber(M) ;; c. h7 S+ ?1 h8 n; M4 n$ @+ J
    % `2 O0 W! z( M) d& b! Z4 M. f0 T
    QuadraticClassGroupTwoPart(Q5);( ^5 S" B% |# d# b1 U8 Z
    QuadraticClassGroupTwoPart(M);
    ( T: ]6 p  }) Z2 N& R! \NormEquation(Q5, -5) ;
    + T/ T- M& R. t$ m+ H, _NormEquation(M, -5) ;
    9 T, }+ R/ s8 h3 L* ]Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field9 ~" ^$ W) J2 N, B% v- g
    Univariate Polynomial Ring in w over Q5+ c( ^, ^: j$ U$ b
    Equation Order of conductor 1 in Q54 B5 z, c& G, {7 B  y
    Maximal Equation Order of Q5
    8 ~# M; ]- V6 BQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    ! S  B8 P" v+ ]- A% {7 hOrder of conductor 625888888 in Q5
    ( W! H7 q4 g" }% v8 Strue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field* @; @0 n7 A( t! n
    true Maximal Equation Order of Q5: S7 P: y& C+ G# j
    true Order of conductor 1 in Q5
    8 ]- p; L  f+ ~' V% u* l& Y5 {true Order of conductor 1 in Q5, O4 I# ~: v! h& i/ i, U4 X# Z
    true Order of conductor 1 in Q5$ G" k! \" n. l0 W& d
    [
    1 A% u1 z3 `2 u# _+ ]' r: p2 i    <w - Q5.1, 1>,6 I- S" }- w  ?0 W% D- l, Q2 _. Y
        <w + Q5.1, 1>
    3 v$ K# p) M8 M]' \3 W) D. U3 |; E2 Y7 V
    -204 d6 q# d$ m6 K/ e7 O
    $ e3 r! e& q( c/ @- x
    >> FundamentalUnit(Q5) ;
    5 E* n7 L- H% G6 S                  ^
    * \5 p2 U. _# t! ORuntime error in 'FundamentalUnit': Field must have positive discriminant. e+ B4 B& B8 `, E% ~

    3 B8 R+ y! _" z( C' Y! x- F( a* h
    % C& `6 n$ X7 L9 P& |2 M8 b>> FundamentalUnit(M);
    & `# o& t/ O' z0 [                  ^
    . s$ _. l. S, {Runtime error in 'FundamentalUnit': Field must have positive discriminant4 m: p. `) s! s3 v
    " i! f4 q) I& B( e
    20
    # N# ~! X: L* B( b  ], |
    " m, e' Y6 a/ b- l) C' D* {>> Name(M, -5);# W4 G5 q4 r8 u8 \3 C) V) e) J/ E$ ?
           ^
    / N' Z+ w: n! w  MRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]/ J7 ?# q8 U' T; v' W* U

    1 c2 }2 `% Q4 }7 `8 E' z% L1 {1( R; U1 D: C$ A* m* W1 }
    Abelian Group isomorphic to Z/2
    ! S+ r7 g8 P: j2 U" n! DDefined on 1 generator2 |% Z( H6 e1 E/ W
    Relations:
    + e+ P" [$ J7 h    2*$.1 = 02 h7 n# ~7 W5 ]+ A
    Mapping from: Abelian Group isomorphic to Z/2/ H) h9 b4 H) t& Q
    Defined on 1 generator
    + I6 m9 ^1 o8 X: zRelations:* M1 b: a: F/ |* G
        2*$.1 = 0 to Set of ideals of M& b9 Z' W8 I3 a$ @
    Abelian Group isomorphic to Z/2
    * y7 C9 ^" r$ ^; SDefined on 1 generator9 Y1 ?# {$ t! @
    Relations:9 D6 R& B, l, K5 Z% H' a' d3 i& k- W
        2*$.1 = 0
    , u5 I/ D* s" K3 tMapping from: Abelian Group isomorphic to Z/2% D$ c# s; {0 V* A2 e
    Defined on 1 generator
    8 |5 J8 \- J  L  P0 J$ _$ \8 J, [4 [& X0 gRelations:6 N& [/ t: |3 Q9 Y3 Q& T
        2*$.1 = 0 to Set of ideals of M
    ' N; X, W5 N) T  Q. r2 {2
    0 \- I  A6 c% L7 e* O26 G# n; g0 k4 Y( z6 M
    Abelian Group isomorphic to Z/2+ C6 d3 F0 K/ E4 t; q8 u
    Defined on 1 generator
    * O. F% @6 Q) E# u/ hRelations:$ J$ q2 A5 ~# L4 c
        2*$.1 = 06 j5 l1 _4 ?' {
    Mapping from: Abelian Group isomorphic to Z/2
    # x$ ]. A+ ~+ `# j# @8 gDefined on 1 generator" M: R1 n, ^! y' V2 e2 u) h. A4 a
    Relations:
    * W  s* g3 \  w. F2 l    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]& q# h  O& p' d& T* `- t3 @9 P
    2& u; T- @' X/ `9 ~- M
    Abelian Group isomorphic to Z/2
    * L- K8 D5 z& L1 t+ UDefined on 1 generator  Q* p3 U6 Y0 V2 K! A' Y5 r+ e4 I4 D
    Relations:' o8 ?0 j4 N9 c$ k  b8 C7 r2 D
        2*$.1 = 01 D! m1 X* O% |- j
    Mapping from: Abelian Group isomorphic to Z/2" T; `4 z* c/ ~8 |, [2 p2 C
    Defined on 1 generator  S5 H& o0 H" C0 N  p7 e
    Relations:
    ( _! Z  F! p3 t1 P% @    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
    ; p! T- l' [7 I4 d  X  cinverse]
    6 F2 i( r' G6 ?7 _3 I( s$ SAbelian Group isomorphic to Z/2/ M  g5 g. z3 ~0 P# {# S! `
    Defined on 1 generator8 A2 R1 G8 Q/ P
    Relations:5 c! I4 P& b2 Y8 l4 f5 L* h7 U
        2*$.1 = 0
    2 N  a( e" }2 Z& ?- dMapping from: Abelian Group isomorphic to Z/2( c6 m$ N, A7 o
    Defined on 1 generator
    - M: j! F, s6 uRelations:
    / I8 w/ d$ Z& h    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no 7 a1 m- J! D, z3 b, Y
    inverse]+ O9 o8 r8 A* ~+ v
    false, X: w3 e  P9 p) F
    false9 O7 R* V6 [4 }$ _- L$ T
    ==============* b  `8 E0 |  d4 O. Q

    # b! V' ]7 q% I( I, ~1 [$ J& b8 k) n1 O) J2 E$ x) `
    Q5:=QuadraticField(-50) ;: c" o/ B! m+ ]7 d, M5 ]1 M
    Q5;
    6 r' S; G% ~' T# Z5 e- T  W' U2 h' T; K6 P* k! B3 t' E
    Q<w> :=PolynomialRing(Q5);Q;0 ~* f& C, N1 S/ K/ o
    EquationOrder(Q5);
    1 {# V8 F; G. H( qM:=MaximalOrder(Q5) ;5 L9 b  w: x% m8 T! v
    M;* C3 e/ I& c1 F" t5 [
    NumberField(M);
    2 d8 A6 y5 b7 r  aS1:=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;
    - A4 u! Z3 _$ f% b9 @IsQuadratic(Q5);
    % ]% |0 T* L8 }IsQuadratic(S1);
    # |3 e" Y! f* X) ]- yIsQuadratic(S4);
    6 @) a; c* z; C6 YIsQuadratic(S25);9 z7 N; A+ O; v" j
    IsQuadratic(S625888888);1 \$ v) s, V7 V7 s
    Factorization(w^2+50);  1 r5 W+ `% N9 C. y
    Discriminant(Q5) ;- b7 p# R! m2 y" p) `
    FundamentalUnit(Q5) ;
    ( c, w" i; d& ]4 \9 H( b7 F9 j+ gFundamentalUnit(M);
    ( h4 ^! c* u5 l: CConductor(Q5) ;
    7 U, p/ f8 u) T3 I4 q4 X3 |0 ?& ^+ A) n  X
    Name(M, -50);
      N: u( c; y0 q. dConductor(M);
    3 X& O. {. U) e0 MClassGroup(Q5) ;
    5 D, |/ q) g2 O" b3 aClassGroup(M);" N; y' R: ~! I# E0 n4 X; f
    ClassNumber(Q5) ;% Q/ ?! }* f. X" p
    ClassNumber(M) ;5 q4 i7 I1 t- B' R/ F0 |$ b
    PicardGroup(M) ;# E$ u8 ?2 A7 J/ B7 Z5 `
    PicardNumber(M) ;3 S, s2 z, a6 Q: G* _& _$ F

    / u3 b0 J& Q1 D) b7 P8 MQuadraticClassGroupTwoPart(Q5);
    ( \5 l' e" c3 h; xQuadraticClassGroupTwoPart(M);& u. C- T7 ~2 z
    NormEquation(Q5, -50) ;
    + c7 i" {) r  ?NormEquation(M, -50) ;" s) A9 A1 `; g( z
    * H4 `" S8 g2 e; u. q  j  Y" H8 {
    Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    " s. G6 f4 [( ]Univariate Polynomial Ring in w over Q5/ Y; e9 f! w: t& e8 h+ h
    Equation Order of conductor 1 in Q5. a* L' `/ v  `2 R2 ^6 i7 V
    Maximal Equation Order of Q5
    ) r( r/ |; m/ n3 Z% B8 FQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    , c9 I: H* J2 L4 M. SOrder of conductor 625888888 in Q5* j) W$ Q. @& K9 |' W7 R4 C- k
    true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    * R9 M! r' k$ p5 h5 q2 Strue Maximal Equation Order of Q5
    6 T0 r, K7 W# o! F% Htrue Order of conductor 1 in Q5( a6 F2 ^/ m8 y* U0 B0 f
    true Order of conductor 1 in Q5/ u" C$ ?; ]: n. e
    true Order of conductor 1 in Q56 u! x1 _+ r0 n, q
    [
    # }' J% c3 e2 U+ s. {    <w - 5*Q5.1, 1>,
      d$ B. E! ?* f2 ?$ p    <w + 5*Q5.1, 1>
    $ c  O4 B% W0 I5 b  A% n4 T$ c]
    1 {% k* R* k7 ?* m2 z% e7 Q" {-8
    ' x4 O" u9 R( Z7 u9 u/ u" m$ |2 T3 Q7 r  M. v7 L
    >> FundamentalUnit(Q5) ;
    7 }( ?# C$ D) J7 Z3 t# M* k& P- J# e                  ^
    6 _5 Q/ I8 y$ h6 WRuntime error in 'FundamentalUnit': Field must have positive discriminant! Y3 j7 }* Q; E* [

    8 ?" p. k  a! x0 A
    1 g  u; Q6 p# S>> FundamentalUnit(M);& K7 t) _1 t+ U
                      ^; M7 s$ b" A& b7 z' k
    Runtime error in 'FundamentalUnit': Field must have positive discriminant3 O7 N( R; q- _& Y! t( k

    / `3 `& }: M$ I; z' t8- }7 i- R8 M$ A4 X, s* n5 r. \
    2 O; ~7 F* {6 Q& e+ Y
    >> Name(M, -50);0 S, a. h: _3 |  L( K* ?7 }
           ^
    . }' S( d7 I' qRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
    " c* E4 L, A/ k2 d
    # H8 N! ~1 ~7 e$ x8 e* A- S3 O5 c# J1
    " A, U3 n" A( j  @( o2 LAbelian Group of order 1+ n3 Q. S$ h$ R' n7 b
    Mapping from: Abelian Group of order 1 to Set of ideals of M+ s5 u+ K1 N9 R/ c
    Abelian Group of order 1
      }/ T) Q7 R* v' k7 n8 x- E' c# lMapping from: Abelian Group of order 1 to Set of ideals of M' B: K8 V: }9 `2 I0 f1 _
    1
    " A) S. d7 I9 y4 N9 s$ C$ _1/ C. C4 q, E" O$ {% @6 Y8 B& ~
    Abelian Group of order 1
    : F' e5 i% A+ e/ @9 @1 _. N* n) f- CMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no* w( i2 d# r" W. w5 F4 |
    inverse]6 N3 a2 J9 K: q  w- X
    1$ M2 d8 Z/ g5 Q% W
    Abelian Group of order 1
    3 M  q) E0 A) o4 i8 g1 }1 s+ tMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant" H  y  |( u1 b) `) T- B, ^
    -8 given by a rule [no inverse]
    # J) x+ ^  {/ d2 c. k: I* HAbelian Group of order 13 X" I' L2 R* l" ^5 _; H
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant3 j( J3 \- A4 ]" P6 P
    -8 given by a rule [no inverse]% C* v# l% e5 D/ O3 O/ t
    false
    " [8 x9 {1 x0 d% `/ k2 {- p! Ifalse, t& g# [# I0 h% ]9 ~& U
    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：, E- a$ r. n( R6 K. R8 V8 e

    * W8 k7 B9 H' u! b, WQ5:=QuadraticField(-1) ;" u: R$ A0 d) ]) l
    Q5;9 D' k" U# S! Y

    " ^/ i* i+ ~2 `. m3 F) q# pQ<w> :=PolynomialRing(Q5);Q;4 b4 `! @3 v- D, ]) `
    EquationOrder(Q5);
    7 J8 @4 S$ h: }6 A0 m  HM:=MaximalOrder(Q5) ;
    & M, S- X+ e, P. f4 G+ Z; }M;+ d0 Q. S7 \& T; c% T5 Z4 W
    NumberField(M);
    ) n4 X. t7 i0 x. q0 v) x. bS1:=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;
    % }' a6 L/ r. j  B% r4 d: U4 ~IsQuadratic(Q5);
    8 u5 e0 B1 K$ A0 z) M/ v0 ]# s$ rIsQuadratic(S1);
      G, n# O- G6 }IsQuadratic(S4);
    & p9 e* z  Q# C" V! I' r% G2 tIsQuadratic(S25);
    $ H6 |/ y% T$ f1 i6 |& oIsQuadratic(S625888888);
    & U  ^9 B( p. H$ cFactorization(w^2+1);  & Q1 k  ~1 a1 e5 L% S
    Discriminant(Q5) ;
    : f) a7 |7 V. r; HFundamentalUnit(Q5) ;
    + [  |, k' y) |7 B* i, `) f5 ?FundamentalUnit(M);7 |2 Q6 A3 v, \7 @
    Conductor(Q5) ;
    ; ]6 Y/ A8 X( J. g; c: c  F3 a
    3 e/ d! t, U7 r  M* p3 tName(M, -1);& m' n  P3 Z  O1 G
    Conductor(M);4 Q; U1 f) v& F
    ClassGroup(Q5) ;
    ! g- i' m2 j# H7 T7 [5 _0 DClassGroup(M);1 d& H: s2 R# M5 V, l0 |
    ClassNumber(Q5) ;
    1 {; Q, r6 {5 d5 U! t: A6 `ClassNumber(M) ;
    ; i- x5 Q) X9 p! |( N2 x% _% E' M$ CPicardGroup(M) ;
    / n6 F* D7 [  x5 L0 N- z0 UPicardNumber(M) ;
    0 c, t: l( q! b" R( p4 k: B; }. X) z4 q$ h0 k. @
    QuadraticClassGroupTwoPart(Q5);
    % P- p- I# g2 Y. m3 qQuadraticClassGroupTwoPart(M);
    8 ~4 @& z& D- w% W' dNormEquation(Q5, -1) ;
    ( H& q' \3 R5 c, V1 W$ FNormEquation(M, -1) ;  ~8 S3 V: ?9 i8 [8 o' V, J

    ; |+ e& j1 [9 T" IQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    ! T6 B9 ]- v$ q5 r4 }$ YUnivariate Polynomial Ring in w over Q5
    - w2 k* d( L8 tEquation Order of conductor 1 in Q53 w4 V4 {. v' f0 {; e. U( v
    Maximal Equation Order of Q5
    + E! d/ h7 p+ x0 o9 QQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    & I# }+ Y" Z' l( ^" T* dOrder of conductor 625888888 in Q5
    6 o- {/ w. j/ k: U2 G- Htrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    0 T0 G' W2 B$ H8 etrue Maximal Equation Order of Q5
    ; ~* l* D  K+ g# ]  k. m2 }, `true Order of conductor 1 in Q5
    . S, b& @! A$ w0 {, J6 }* \' N! ?true Order of conductor 1 in Q5
    : F; ~* _/ G; o' }. O  Btrue Order of conductor 1 in Q5! F( `9 V/ c* Y1 g
    [
    1 i2 {  i2 ~4 [+ S    <w - Q5.1, 1>,
    ! L: k" Y$ x7 s. @) Q0 r$ C* n    <w + Q5.1, 1>: k/ ^) j1 G/ D3 S, }
    ]$ e. C7 [7 L) L
    -4& ~5 U$ M  L, e; ?- m$ \

    7 m& Y/ B1 T) \- r7 j$ A>> FundamentalUnit(Q5) ;' u' ^& i$ {. |4 S3 s% R) ^' i9 w) d" }
                      ^
      O0 M( \4 A+ \Runtime error in 'FundamentalUnit': Field must have positive discriminant
    # M: m- Y+ n( V# _* T* l1 T7 c* g' _

    1 r- ^' t" ^% S; o0 c>> FundamentalUnit(M);8 @$ ]0 b$ t$ T9 x" _; v- I) b
                      ^% g) x, J2 i0 e1 L7 I
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    + G2 v/ K$ u3 Z/ N2 x
    ; B/ O; q% F4 l! Z; Z* ]7 g4/ ~9 l1 P2 {: ^4 G6 P

    ; y% U: s3 \) e* b3 L, c7 ]>> Name(M, -1);6 N" j( F% J; K, ^4 {1 ]1 j0 r0 s
           ^# N- Y2 F7 B( N: r+ l. ]: g; I5 y
    Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
    $ R+ h7 ~- I& ]) ]7 N; l; j; |% c' S; u* E
    1
    * e2 L" v9 \) c! }8 F+ YAbelian Group of order 1: z8 F7 o( v2 t6 m9 b. {
    Mapping from: Abelian Group of order 1 to Set of ideals of M9 b) ^- w. ^* k7 X3 K: M
    Abelian Group of order 1, i' [3 \: |, s& ]. e- J
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    8 S5 f* X: ~( t, V, Q; I& u1
    1 [7 I8 \$ _2 ?- u% C+ q1$ m, I" K  x' Z$ A; x( `
    Abelian Group of order 1! g7 q9 a! c' K
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    ' ~4 a% W8 h0 g  O% Q" L8 c! Hinverse]
    . d4 `6 e* O7 d2 h1
    & `% I- m3 ^0 m! p. uAbelian Group of order 1& }. g; H3 \; F+ o7 W; ?$ T
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
      \' x6 F, u- H6 [-4 given by a rule [no inverse]
    ! n) S! e- I9 h' Q' [3 n) a. ~: pAbelian Group of order 1% n! ~7 S! ^, L% g) Y+ v, I
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    " B7 S2 g. T% T5 d3 v* h% x. m-4 given by a rule [no inverse]
    , Z. s% O) V. bfalse
    4 m0 e3 I4 {5 E1 ~! T, R2 Yfalse+ ?, g1 h1 f7 p# ~1 ^7 _4 a
    ===============
    / B$ i& ^3 H, t
    ; F4 p8 s4 E# DQ5:=QuadraticField(-3) ;6 P. h9 W3 [, |6 N) ~4 w
    Q5;
    ) @& K4 C8 ]& P. Z1 c& P0 L: Y# w8 _! z
    Q<w> :=PolynomialRing(Q5);Q;+ c4 q& s5 W& I0 u
    EquationOrder(Q5);
    # \( x( g. `0 ]+ p; y+ q* m( yM:=MaximalOrder(Q5) ;
    " Z, [. `% z* E$ ^; [' v6 d/ d5 c% {M;
    ! ^7 f1 Y- ?/ y/ ]: m% ENumberField(M);
    ( k: @# j$ P( w, C* ?S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;2 o! ?5 J: J0 {6 ], J  f& o
    IsQuadratic(Q5);: J: _' ?6 P& |2 E  n
    IsQuadratic(S1);( s* f- M! U' g3 i- U0 P: h! o
    IsQuadratic(S4);
    4 {) j: Q1 ?/ M+ NIsQuadratic(S25);/ k" Q+ B6 U4 h9 K. Y; U4 l
    IsQuadratic(S625888888);
    % J9 Q, l1 }" \) p4 qFactorization(w^2+3);  
    9 a3 j: U6 z- V* P: WDiscriminant(Q5) ;
    8 c/ t6 P0 e; y6 N% ?FundamentalUnit(Q5) ;
    4 e& l! K" a9 d, v+ DFundamentalUnit(M);& n  d$ \9 x7 Y+ `6 d( X
    Conductor(Q5) ;
    0 ]5 d* g* Y, ?1 D) H6 M/ d" X- P& S* V( r, w" i
    Name(M, -3);
      ?+ {: |/ R* D, m: S; AConductor(M);0 w9 c  Y# l+ A
    ClassGroup(Q5) ;
    5 W" p( W1 D. f# \2 F: \# x/ tClassGroup(M);; _4 y9 x! l# O2 u* `( D9 n
    ClassNumber(Q5) ;
    % T. [7 B1 n+ p- v6 ]$ _: a& p: b  fClassNumber(M) ;
    % |( I( k- p# N8 n/ w& uPicardGroup(M) ;5 a5 N" {2 J& P* Q* q3 c' K
    PicardNumber(M) ;
      H. N5 T( j7 R8 \9 h( v. R, e. y) G1 f" n/ n0 D5 ?( J5 B, s  B
    QuadraticClassGroupTwoPart(Q5);
    " T2 o8 K3 |" o# u6 RQuadraticClassGroupTwoPart(M);& Q- N" |0 V* Q5 @9 k# o
    NormEquation(Q5, -3) ;
    , e5 u5 `4 s9 Q+ W& ZNormEquation(M, -3) ;
    0 s- n' Q) g* Y
    9 ?  a: h- S" g" o4 y# SQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field1 [+ ]# M( N4 j3 L1 P9 m
    Univariate Polynomial Ring in w over Q52 _, ?( Y9 K4 E9 t+ P2 t
    Equation Order of conductor 2 in Q5
    ! y' E4 A/ z% Z7 ]# F# a" Z  c2 OMaximal Order of Q5
    - E! R% x; v) }! i$ s7 ]Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field! u; ?! ~# K" ?( P% i
    Order of conductor 625888888 in Q5
    7 O! @" i7 h. wtrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field/ g- X( w. x2 y: K) {5 e) a; U
    true Maximal Order of Q5( m( l3 c1 A& {0 B
    true Order of conductor 16 in Q57 V( p( _$ a* u
    true Order of conductor 625 in Q54 _2 |% N/ W# `  W
    true Order of conductor 391736900121876544 in Q5
    0 x  c( l% W$ F[( Y. D; y& v, s7 u8 m
        <w - Q5.1, 1>,; J1 V! z3 B$ M) o2 E3 N& Z
        <w + Q5.1, 1>
    9 c/ Y, F( W# m, j5 L]: E  f: R# t! ]6 u. V0 O) z( H
    -3* J" W' \( `! e. x( ^/ R0 ^

    % C( V; m) F( K# h1 G+ z& g& C>> FundamentalUnit(Q5) ;$ C4 p6 k& a& S7 U) q
                      ^
    9 r; T# {8 M7 y1 O3 p8 [Runtime error in 'FundamentalUnit': Field must have positive discriminant
    ) X/ \: Y( L( j( j8 o" s' o0 o$ }4 ^
    . B/ X& ]7 A/ Q$ U; ]* ?& l6 V" ]
    >> FundamentalUnit(M);
    / P& R5 h4 O" `1 C- b0 P2 h) }                  ^1 I& E. u  E/ U6 b0 z
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    8 e# x; z0 k% a, J$ n7 [8 W+ `
    & c7 Q5 p* W( Q. O! I3
      l; i: n2 i' ?0 W; z; b; r- _* N' w8 z
    >> Name(M, -3);+ J# M' M5 y) n; x* D1 V' O
           ^: ]  P& t. y& Z2 e3 |( p( ?4 l
    Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
    ; z) R7 ?# L  f3 f& y8 O
    & y4 f# w3 f9 n: q' ^  _1# r( r% b0 p+ u: T
    Abelian Group of order 1: ^9 m1 @: G& L2 I* o; C# z5 g. R
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    ; H. e5 k5 R( }Abelian Group of order 1
    ( x2 z* M( B' I4 eMapping from: Abelian Group of order 1 to Set of ideals of M
    7 @, h4 N3 B$ }* ~; n1
    # a  y, c. n2 U- w9 b: j13 f' p4 {, I( g2 I1 V) O3 k/ @% W
    Abelian Group of order 1# x  R. |4 J" ]1 }; S4 C+ v3 P$ y6 O0 b
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no% p8 j" g9 {# J1 F  t- O
    inverse]) p% d" @- T$ n. n; ]' ~) I
    1/ T" \5 a6 S/ r+ Y
    Abelian Group of order 17 @3 W, Z* e! M+ [# r
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant$ @7 R3 X  S# e! I( o" ]
    -3 given by a rule [no inverse]9 f5 K3 Z4 \( Q- O# A2 H0 g' g
    Abelian Group of order 1, K  U" P. W) x4 {
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant/ M6 j& L: G" _6 \2 D
    -3 given by a rule [no inverse]
    ' C# H- w+ ~) f; v, X9 hfalse9 P3 F/ Z3 T; v) y+ |
    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

    群组: 小草的客厅

    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
    0 F& Q) K: V8 H9 \% e" Y- d- Z" \4 f% a! O4 Q, q6 m" L
    Dirichlet character
    0 j7 H) |, e2 @Dirichlet class number formula
    2 L2 o% p/ D& L7 b2 Q7 S( t4 y/ B: H2 x6 n
    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0 E( o+ i1 a5 m$ X
    $ @! n2 ^) X! U( b9 L
    -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/ h4 ~( ]% X8 A, {4 e$ Q

    8 ^5 G0 p( T$ G4 m8 ?-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，# V* @7 ^* @' I! B4 z5 F
    h=-6/（2*3）*Σ[1*1+（2*（-1）]=1- D, E  D8 n5 `- }1 U
    2 C/ m! P7 k  x, {- C, ^
    -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，) V  j  D+ U9 I

    - \' ^& Y8 W" u' w3 l2 s0 x+ a' m# z! D6 F

    7 {' F! |' S( a) b- i9 Th=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2- ~& a6 O. i( u- i) G+ S

    ( y3 C; N' V! S: T  }  E3 f$ q# W; E: _2 q- o* f+ b

    : {. |0 y6 b) Q( ]6 v( @* g-50时  个单位根                          N=200
    ( M, I+ Y+ {) a6 E, c
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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: 219)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 227)

    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: 229)

    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
    ) {3 @# {7 }' N
    # M  S8 X2 j  j7 u: ^1 j# GF := QuadraticField(NextPrime(5));: i" k( t5 I6 R2 u& o

    5 o! U# n2 E) X, X' v3 ]5 ?9 {" AKK := QuadraticField(7);KK;
    ' A0 k1 Y% @' X6 sK:=MaximalOrder(KK);
    4 r- ]5 _: k8 `' B& L" MConductor(KK);
    ! ^, l6 e0 ]9 n% K$ O" L# cClassGroup(KK) ;, e5 @& O; F2 m  M1 @$ W
    QuadraticClassGroupTwoPart(KK) ;
    / c5 Y, X9 q- C$ D7 p2 PNormEquation(F, 7);
    - K4 n: x0 f& R# |A:=K!7;A;
      j4 [& G; x2 y* s  _" g) yB:=K!14;B;6 _6 h6 V% P$ m2 d% {) [
    Discriminant(KK)
    & w& f4 e9 C3 r: H
    ) @2 W7 ^0 b, h/ X3 z" ZQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field% C7 y  }) S6 ]3 ~& p2 N
    28
    / q9 C/ j, C1 [, h6 L$ ?Abelian Group of order 19 k( a/ F, M& O$ Y( }
    Mapping from: Abelian Group of order 1 to Set of ideals of K) a  p+ H; X6 I* ]
    Abelian Group isomorphic to Z/27 i$ N9 s9 j- r3 B. j, k
    Defined on 1 generator+ }: O! V. P" k# t# \/ ~) m
    Relations:
    2 c2 k& Y3 o+ Y1 k; O0 E    2*$.1 = 0
      v  M/ H7 e: _& L* i+ k, XMapping from: Abelian Group isomorphic to Z/29 O6 p( e. ]! N; [3 o/ r+ N. G, s
    Defined on 1 generator+ u+ u  p, H7 B
    Relations:
    ( u- r4 p8 l0 T! S7 F% p( k. \8 l    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
    5 m% h, b$ A, M; x. finverse]
    7 F! u7 @, _- {  D- b( gfalse& j9 i/ _/ \: i1 o; C
    7. o; i5 z7 s" u. h- d
    14
    ! Y$ `4 t# X4 }% ^% _+ z28
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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
    . \% e1 [* U6 \8 k! G+ r# d: X$ W5 S6 T
    11.JPG 4 \& n/ p0 k/ v# H8 k' l

    6 k* a1 s& ]9 A3 n8 E7 s. K 3212.JPG
    ; j% W5 h3 D3 p- k( l' g" I  b3 s
    8 J" u; V/ ^1 A+ t8 X 123.JPG , K0 K6 i# g9 e# n. ?

    . X* W2 ?* f: P9 g% Q& O  o分圆域：! C$ `0 U5 K1 e: ?' k
    C:=CyclotomicField(5);C;
    1 H5 ]" `. @% B: x3 C9 [  H3 l( }CyclotomicPolynomial(5);
    0 N  L2 X  S% n$ |" a: |) y. ~) G. D( ?C:=CyclotomicField(6);C;. G/ l$ Y9 O$ O$ z
    CyclotomicPolynomial(6);
    / E0 D1 z) J& Y3 V! |4 RCC:=CyclotomicField(7);CC;
    . ~8 K5 w& R: z: o. E, u+ wCyclotomicPolynomial(7);
    9 @: [$ f* t9 X1 d) }MinimalField(CC!7) ;" }% F! `  r1 S$ p4 B( j0 R- G
    MinimalField(CC!8) ;# B' H& Q0 p  R% R) |; W
    MinimalField(CC!9) ;
      u! q+ L; ?  [/ ^/ ~5 O; pMinimalCyclotomicField(CC!7) ;
    1 Q; h) I" B- v" T8 V+ G7 N, nRootOfUnity(11);RootOfUnity(111);
    " Y5 ^8 q8 ?: W% y$ PMinimise(CC!123);4 e0 ^: r0 ]& x) d* Y* W
    Conductor(CC) ;$ H8 o9 {- ^, y1 x
    CyclotomicOrder(CC) ;( ^7 I' s" e& D1 }
    8 u+ U; M( D( B/ g! q
    CyclotomicAutomorphismGroup(CC) ;
      K$ g/ f5 c. E2 r- ]' e: E# h- O9 T* Q" h
    Cyclotomic Field of order 5 and degree 47 T5 R( W5 M) [9 m0 z# _; ^* Y3 P
    $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    7 p6 Y6 w, z& m; b4 y: ^Cyclotomic Field of order 6 and degree 2' j2 `  ]' R% N# A
    $.1^2 - $.1 + 16 A$ s  w) p2 k) U; F# n; ?3 ?
    Cyclotomic Field of order 7 and degree 6
    " k+ U' j. D2 [/ S! C$ n/ E" ~$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1$ ]8 ~4 N8 j$ S% p$ B( c
    Rational Field
    6 K# g& V' D3 R5 O, |Rational Field% Z& L0 v% l& l, R1 }
    Rational Field
    $ W* I2 v8 D2 k' A) V( VRational Field
    $ g. N; g$ I) ]) B7 o% t% E6 Wzeta_11
    ! r6 E7 s' W! |1 Mzeta_111
    * N! h4 C& B9 L* b2 |& J. H123
    + N8 R. V9 f% w' E0 ~" Y7+ \8 L( V; F/ V( |/ v
    7. `" `$ R- g5 C5 C8 {
    Permutation group acting on a set of cardinality 6) ~$ @6 s1 y  B' \
    Order = 6 = 2 * 3! y$ y& R% P- n1 k
        (1, 2)(3, 5)(4, 6)
    ' k! Y; g7 l- ?% e4 J    (1, 3, 6, 2, 5, 4)
    - N! Q( u. R! v3 R# f- CMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of 1 P/ K' Y6 W, r/ i7 {. ^" U4 B
    CC% }8 ^( N* F; p: E& x0 o4 ]
    Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, 9 `5 @7 n7 L. B9 i
    Degree 6, Order 2 * 3 and' l8 |5 g5 B4 C; ?  p
    Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    ( I, P5 @0 b: I" H* ]& `6 N) K3 ACC
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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
    & R' J0 O( m8 R. W# F0 m! E
    lilianjie 发表于 2012-1-9 20:44 + S6 b( Y6 H3 W, U
    分圆域：
    0 x  B+ k& \1 }  y. i) g6 xC:=CyclotomicField(5);C;( d* g; R7 w3 i- {9 G
    CyclotomicPolynomial(5);

    / K/ a- B& x6 d- _0 l# D0 Z( T5 z( G7 _& w" j
    分圆域：
    ( J7 f' W& w- s分圆域：123
    6 r0 ?+ M& ^* V8 |0 {3 L5 I
    # C5 v- ^$ q/ n1 _7 G1 \3 p. f' zR.<x> = Q[]
    0 z! }, I# i; u; FF8 = factor(x^8 - 1)8 O7 F! e$ u! ]8 k
    F88 O- w% g1 o8 h
    % W- y2 F$ J$ S0 e5 B8 X6 \6 b+ v
    (x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
    / S* I" E8 a& k4 R6 Z1 B5 [% L5 ?/ J, H% @5 S$ c2 R
    Q<x> := QuadraticField(8);Q;
    : x2 q2 I5 _* C6 r2 R" RC:=CyclotomicField(8);C;( X9 G: E# c8 o
    FF:=CyclotomicPolynomial(8);FF;
    # b! p* G) n6 u: ?7 s; B
    ! f0 \" u: ^, j3 t% MF := QuadraticField(8);
    * e- B1 w9 B/ u; C/ f0 ?1 f4 PF;
    8 R1 c+ ^( v3 K" x$ L% I: qD:=Factorization(FF) ;D;
    9 p6 b8 G& L$ iQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field6 r  ?0 e$ }7 k  d7 v
    Cyclotomic Field of order 8 and degree 4
    : a% X7 O- @, \4 q" N$.1^4 + 1' `$ U! V$ q# _0 ]' b* I# [7 `
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    ' G3 _: n0 ~% j[
    $ P2 h' f* D3 U0 f! u6 Y    <$.1^4 + 1, 1>
    % n1 ]1 H5 q; v- U1 q]
    6 N* w+ I8 `0 }: J' U$ {' m1 c  _5 {% L, `# l  K
    R.<x> = QQ[], G" x( e0 p6 @; z% _8 P5 a
    F6 = factor(x^6 - 1), t) h# l+ q7 T, {8 b
    F6
    , W) k+ `/ }5 h. F, q- T1 G* U7 |: x* y: k
    (x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
    # R, d9 H- P! C1 _5 m
    4 K: Z. n/ _1 {3 r% a3 W& Y+ dQ<x> := QuadraticField(6);Q;6 S7 Z  q) W' f* Q, n
    C:=CyclotomicField(6);C;
    ! J+ q+ ^/ F6 t  Y% QFF:=CyclotomicPolynomial(6);FF;; M8 y) _6 ?. f4 ?  g5 f

    2 u$ |- Q* D$ l) RF := QuadraticField(6);# ?5 K9 `% g# _( u. @1 v7 a
    F;
    / X+ {. j/ w+ pD:=Factorization(FF) ;D;
    7 i. F# z9 x$ p7 v- E" z4 ?Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    & U8 e# t, Z$ Z$ v* n, s7 J* V3 i9 YCyclotomic Field of order 6 and degree 2- H. ~7 f' {: Z! m
    $.1^2 - $.1 + 1/ o# R6 h8 u& L- j9 e: t  V- k0 B! X
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    0 m6 H: |. ?) q& r: e3 Z[+ {' s3 d* T, t$ a! A
        <$.1^2 - $.1 + 1, 1>
    ! z& W* `7 E: U- t]
    * X, b$ K, K9 F$ \
    5 C# j4 i2 q' n5 f* {$ uR.<x> = QQ[]
    ) O, R  k0 Q, s' J! |( Q& DF5 = factor(x^10 - 1)' b. H$ f* q4 v. ]! ~! }; r* g& r
    F5! A$ Q! s: N& m
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +3 E5 B& o. P) p+ ~
    1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1). ^8 q$ r. b2 e+ B# L

    - C) l3 y  J3 F, }9 }' iQ<x> := QuadraticField(10);Q;2 d4 w3 v8 v2 g5 ~  l; Z8 g
    C:=CyclotomicField(10);C;! K$ m! t: P4 `
    FF:=CyclotomicPolynomial(10);FF;6 s5 q0 R- W3 x
    ' z8 p( E7 X; ?
    F := QuadraticField(10);" [% e. L6 S( ^4 C. J
    F;
    , s5 z' l. f% c5 p7 RD:=Factorization(FF) ;D;
    . ^7 [! y2 q0 Q( Q, X/ TQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    ! [1 Y- v; T" H, ?2 }+ MCyclotomic Field of order 10 and degree 4+ p! X0 @2 u& F
    $.1^4 - $.1^3 + $.1^2 - $.1 + 1" s' r( g4 c( H& D( t1 b& `
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field3 b& L' q8 L3 V
    [
    ' T. V9 m) L- V) j% w* m2 n$ r6 x    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    $ d" g- A- I5 g5 \1 L* b]

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    2012-1-13 11:49

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