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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
    5 @, G4 I+ T# V% o! n# Y
    ; W8 `) e  a# P6 [Q5:=QuadraticField(-5) ;
    ( b2 C4 W7 Q" H) u6 aQ5;
    % ?6 k2 f! z# V8 @  j* H$ H) ^
    # B' S! K: H% l+ R( ZQ<w> :=PolynomialRing(Q5);Q;" q: K1 V1 @/ B; e' X" q+ o
    EquationOrder(Q5);
    9 e% s  {- x5 E' o; JM:=MaximalOrder(Q5) ;
    ( Y, e0 E0 n% W* S" H# Z+ oM;' e3 r- N) c2 N2 q
    NumberField(M);
    8 d7 {1 [1 P, i' o9 {2 w6 W! m8 IS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
    . k! e1 n% A. e8 _' {% ~IsQuadratic(Q5);
    ; ?8 Z) Z* X6 v5 q: E1 T$ \IsQuadratic(S1);2 n& J) {4 T0 M1 m2 d5 Z
    IsQuadratic(S4);) Y  S5 {* V+ W7 d& I, t
    IsQuadratic(S25);
    $ ~; q0 g+ L0 e2 c% W  N' V% q5 r8 P+ fIsQuadratic(S625888888);
    & C0 n0 i0 b6 p& O1 d# tFactorization(w^2+5);  
    0 t- {" {& b% o. FDiscriminant(Q5) ;" ?) g1 t! V% _
    FundamentalUnit(Q5) ;
    , J* {. G7 S, i* ^FundamentalUnit(M);$ Q7 f3 A- P. L3 X% }" |
    Conductor(Q5) ;
      w. g/ y$ P2 d) m0 \4 i9 T+ \( }5 C' A, m" h& {$ @: P
    Name(M, -5);" d) e" l, M' @! a, V8 @
    Conductor(M);
    ' ?" B( }* p& FClassGroup(Q5) ; 4 Q2 G* z% {& [8 m4 E. [
    ClassGroup(M);. C5 \+ F  T/ _. `
    ClassNumber(Q5) ;
    3 _. x% v+ b0 P0 `ClassNumber(M) ;+ }& ?2 C4 H. q. U/ C' O
    PicardGroup(M) ;
    + z7 B' I8 L, {2 U  nPicardNumber(M) ;
    / F& \! F7 g) @4 y6 t! Y# I; n# r+ x8 |4 X4 J# f5 m
    QuadraticClassGroupTwoPart(Q5);; `' y& U  C8 T6 e% z/ V
    QuadraticClassGroupTwoPart(M);
    5 g% x; D3 m  J/ k0 yNormEquation(Q5, -5) ;; A$ u- Y$ \: P$ r6 u: i3 k
    NormEquation(M, -5) ;; x* p! U- O8 C
    Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    $ A- N+ T# [4 s3 e$ T) jUnivariate Polynomial Ring in w over Q5' f5 F: J) h9 |* U: ~" a
    Equation Order of conductor 1 in Q5- I" @  w1 h2 x2 c4 }3 D
    Maximal Equation Order of Q5
    : T  h0 ]6 ?2 y) }Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    " |/ t% @' o6 o* }( E; VOrder of conductor 625888888 in Q50 }9 l$ {3 U3 k; h% _: d
    true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field. ~+ X; d% J$ I& C8 M' q
    true Maximal Equation Order of Q5  u  _  D9 ^5 n  t/ ?: O8 F
    true Order of conductor 1 in Q5! x4 H( q8 V5 n  u9 }
    true Order of conductor 1 in Q5( G3 t( L  X- [
    true Order of conductor 1 in Q5- ?' Y2 w  F& O* x% g
    [5 C/ t& l2 @8 z0 g7 U) Y
        <w - Q5.1, 1>,
    ; A' W/ ]: S8 S$ e0 k! u' J    <w + Q5.1, 1>
    ) b3 j0 D& M5 ?]
    " [( u" z1 @5 W3 ?-208 I1 R! t7 o; R+ W: X
    : ^, v# N9 w  P. r3 H
    >> FundamentalUnit(Q5) ;4 ]& X" g; t( p. g6 C
                      ^8 v+ G& J" F" j* E& g; D
    Runtime error in 'FundamentalUnit': Field must have positive discriminant' B% E, v, E$ e

    % ?6 e! `6 V0 l  r! e5 i5 \' M2 Q9 j8 P: e
    >> FundamentalUnit(M);
    + u2 _3 C; ^- x2 n                  ^
    5 R! m1 K$ }4 C1 f! g) F2 QRuntime error in 'FundamentalUnit': Field must have positive discriminant) U6 f1 m" g2 a: U

    ) p1 G' M" n; b; u20
    & t% _  R$ }9 T# [& {$ M+ r) S$ w) s. [
    >> Name(M, -5);' ?, B' ~' t+ @/ f# w. e/ b4 q. d
           ^& X( z( F' w+ j8 ?  k
    Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]8 w. R7 r; K/ E& T, m5 [
    ; w: p2 T  g% S0 c2 c1 t+ q6 N  C
    1
    / k1 e, H% o; c! k0 o6 {4 WAbelian Group isomorphic to Z/2  U" T& S: W+ S  l: U1 m
    Defined on 1 generator4 E% e( u* }9 P. d3 c
    Relations:
    , P2 D$ `' Y: |    2*$.1 = 0. D$ {! q$ K9 C
    Mapping from: Abelian Group isomorphic to Z/2
    # n# [+ H2 `* VDefined on 1 generator! ]7 Z6 w) w7 p9 e) A
    Relations:
    " S6 G% J' I* d4 B* A3 p; ^. [: B    2*$.1 = 0 to Set of ideals of M
    & p9 O1 L* m; FAbelian Group isomorphic to Z/2! }* j/ |# N/ X7 {/ X) I
    Defined on 1 generator6 J) Q* S4 E6 I! |/ C" v- G7 B3 w5 N
    Relations:
    6 W% b1 c: m8 S; b* c" j+ Z: K    2*$.1 = 0$ s0 Y# C" F' Z6 |8 R0 C
    Mapping from: Abelian Group isomorphic to Z/2: U% D; J# W7 {  u) e1 M, ?: O  J
    Defined on 1 generator
    . @: g6 c6 t+ s9 G: rRelations:9 F" ]% O* ^( _# C4 A5 d  E
        2*$.1 = 0 to Set of ideals of M& C; g' _; r: X
    2: y. X  L# h" J1 }' i, u
    27 A# y: I: Q. N
    Abelian Group isomorphic to Z/23 ]. s' \* Q( a; J: T. B( t$ o
    Defined on 1 generator# w- z+ T6 M7 E
    Relations:+ m; f" D5 g" B* o
        2*$.1 = 0
    ) A' d% [. h6 j; i7 D3 C$ IMapping from: Abelian Group isomorphic to Z/2
    # T5 J, E3 s0 I/ b$ g' O: xDefined on 1 generator
    1 v0 m8 S% p* m; L: JRelations:
    $ N2 R8 A) M, r& e7 k& M    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
    ' E& C0 p, T5 [6 I; X# G. Q22 y2 M6 }4 b1 h" ]
    Abelian Group isomorphic to Z/2# s; [. A/ E' Z9 H# r4 t
    Defined on 1 generator! B$ O9 \8 }& k4 t0 r' \# l
    Relations:: `# U+ f  r0 A: c3 H
        2*$.1 = 0
    9 X) w2 f' a! c" PMapping from: Abelian Group isomorphic to Z/2
    ; K( L7 q5 }! }9 s/ u7 TDefined on 1 generator4 c# C/ l& D9 }- [# Y% M$ y
    Relations:
    : Y8 |' C2 i( k    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no ; x3 Z: x( \7 W+ z0 `
    inverse]7 D3 d. }/ L# Z0 Z0 K1 g
    Abelian Group isomorphic to Z/2' h% K  Q; @5 Z' ~5 Y% H
    Defined on 1 generator
    3 x/ P+ A  Z8 ^! s2 F: W) {& V- PRelations:
    0 D8 i( J7 R5 `4 l& v, X    2*$.1 = 0
    3 f% V* @+ a! ?3 x4 ]Mapping from: Abelian Group isomorphic to Z/2
    2 |. N* d& l" v/ R! v4 \7 U/ YDefined on 1 generator
    4 M5 u& f3 t  y+ X: n, U8 ERelations:
    . R5 _$ D% [% ]    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no / s1 m  H: R& C& C. b
    inverse]
    0 v3 ^# }; U, m# o" S, Gfalse
    % J- k' f0 }6 F' Ufalse
    4 N) x& @7 o' {4 B% X==============
      h' a. d3 O; G3 ?5 }4 a# B3 H+ s9 F

    4 [5 P; k# D: F- I7 q* O5 [$ C6 g9 ^) PQ5:=QuadraticField(-50) ;
    ( ?/ D, s0 w3 ?. @3 p; Z, f% fQ5;
    ; ?% y$ }3 r! {5 u7 E- I$ S
    6 h# w$ ?/ A5 z" C( U" A6 N8 j$ EQ<w> :=PolynomialRing(Q5);Q;
    6 }0 s8 v9 s8 p! Q# Y6 H. FEquationOrder(Q5);1 g/ V3 s: V& R1 `4 h- B
    M:=MaximalOrder(Q5) ;
    , k( L4 `  k5 U8 D2 UM;
    & a% W. F! A/ eNumberField(M);
    . `. x, d5 ~& _: O& kS1:=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;
    6 p7 f( g+ @1 e+ H3 f, mIsQuadratic(Q5);
    # g$ P8 Q3 {' i+ b# Y. ^6 u2 i0 VIsQuadratic(S1);1 S7 W. s6 Y! Q9 |: h
    IsQuadratic(S4);5 t4 }/ y* P' Z: p. y$ n
    IsQuadratic(S25);
    % H2 T; L$ f7 f) O# WIsQuadratic(S625888888);# X0 Y* D/ F' x9 ?+ R+ C! a: e
    Factorization(w^2+50);  7 f) ]3 N6 e, K7 Y( O
    Discriminant(Q5) ;* g- |  _7 K8 o4 B+ q8 I" b$ y
    FundamentalUnit(Q5) ;
    4 d0 e4 l" O8 ?  L) jFundamentalUnit(M);
    ' d2 q! U. j) g: ?Conductor(Q5) ;4 w: g& Z) V& A
    ) `! c* Z: ?5 P4 _8 Q2 F" N
    Name(M, -50);
    0 C5 n9 S4 g0 wConductor(M);: g8 o  m6 Y* h8 B/ O8 [4 A! J- l
    ClassGroup(Q5) ; ' h0 J- I8 T( l
    ClassGroup(M);
    ; z8 r6 Z6 w+ G& t; X# x# XClassNumber(Q5) ;
    ! e9 D7 o* u! g  R+ ?ClassNumber(M) ;
    . a+ X) `2 J* y  L" IPicardGroup(M) ;
    * N. `$ }! I7 {* Q3 A$ t& ]PicardNumber(M) ;
    0 s+ n# b$ O# [9 c
    2 L( z9 t) c- H' s+ z. P# AQuadraticClassGroupTwoPart(Q5);8 r6 X/ T% @* e9 a3 P5 R6 l5 F% a
    QuadraticClassGroupTwoPart(M);
    ; y- f3 d& @; L- S; g5 UNormEquation(Q5, -50) ;
    ) _+ o9 @9 v% N: ~5 a1 {NormEquation(M, -50) ;
    . h) t3 p' x: B1 a
    . ~& G* p4 I  [$ PQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    ( |0 z9 `7 \) [Univariate Polynomial Ring in w over Q5
    % j+ O- ~/ ?  T) BEquation Order of conductor 1 in Q5
    & _0 E: o" l1 w7 x# u7 P3 m: b; ^* H- F/ RMaximal Equation Order of Q53 h( @+ H" z) V
    Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    : _/ o0 y3 T/ V. QOrder of conductor 625888888 in Q56 l4 E( P! B0 m0 j; ]+ {( Y; v/ o
    true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field1 H1 k7 H" G3 n2 e" c1 C
    true Maximal Equation Order of Q5; o2 ]0 K0 [) c. S5 X; x3 d
    true Order of conductor 1 in Q58 W3 Q( |+ t% ~5 Z
    true Order of conductor 1 in Q5
      |4 }; n: d% _5 c0 `; i) Ctrue Order of conductor 1 in Q58 D. h( ~: B' {6 s" s" P8 o. T$ ~
    [
    / U6 x1 j2 H% R* C    <w - 5*Q5.1, 1>,9 A9 v% p% S4 Y% F+ E
        <w + 5*Q5.1, 1>7 p1 x! v6 p5 `3 k: q
    ]4 j$ J  |( C: |/ O! F+ w
    -8
    ! ~) q- e1 s# T$ j
    2 V) y9 @9 Q3 w" ^. Y4 r) B>> FundamentalUnit(Q5) ;
    . \' T2 T& b( ~* n- I- B9 |                  ^
    ( w# R, R0 t7 n$ yRuntime error in 'FundamentalUnit': Field must have positive discriminant
    + g2 n' F8 G( y- d7 q# \3 n9 _% r9 R* ^9 V3 }& G. x/ a1 B
    : j# m. W+ b! {3 N+ r& T5 Q1 q
    >> FundamentalUnit(M);3 T2 R) f+ V4 O& l' Y& _6 u: g, Q
                      ^
    . c9 W$ g# e. B6 |# ORuntime error in 'FundamentalUnit': Field must have positive discriminant8 w: a3 s& D% Y' ?$ K, K8 a
    3 U7 n6 R! b) J$ x6 q3 }2 V
    8
    $ i$ D2 e( d  t  U# o% }% L3 l/ n& C% F
    >> Name(M, -50);
    0 y# \9 X! D6 |9 R* B' D       ^
    . }5 j8 O3 n$ K  w# E% Q% F$ kRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]0 }8 |7 X. V  T% B$ I
    / v  N* V/ X8 H5 z- G* W& ~
    19 c0 `6 H3 w4 Z
    Abelian Group of order 1
    $ u6 Q) N, `% X7 e: [/ W: \+ IMapping from: Abelian Group of order 1 to Set of ideals of M' x$ v' N$ _( c, E, g
    Abelian Group of order 1( h6 ^6 B5 c# B$ f- x2 @
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    . s6 ]2 v0 ^: {* i1
    / ]/ H% X( V5 t2 d/ v1 v2 {1
    / s; p0 [0 `9 b* F# Q3 _- ]Abelian Group of order 1; p8 W6 j2 j. C+ j
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    ( F- \! P' J9 u# [' f' w4 Y  Binverse]
    * A- e0 h6 P  x- J: z9 o6 v11 @0 D9 a2 ?% P; P
    Abelian Group of order 1) e  M- A3 a5 F
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant  W9 K* z* x& V+ Z, {* o$ e5 C2 I
    -8 given by a rule [no inverse]8 V+ q2 o2 k' L. z0 p* ^
    Abelian Group of order 1
    3 ?* X' o2 ^5 Y7 v/ L. b/ U$ wMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    7 C. F; H' d) R-8 given by a rule [no inverse]
    5 [4 O# N" O+ T1 ifalse: C; g9 s$ D5 u. t
    false
    7 u" Z( k* k% \+ b8 D6 _2 ?
    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：
    ; B1 f* L$ G7 J% N: _# ?/ N
    & i+ V" R! E: B3 `  b  qQ5:=QuadraticField(-1) ;9 r* R# _) i( g! _. }6 \! E1 l
    Q5;' K1 j" ~% K/ N2 c6 S

    , y% [8 y+ U, j2 ^6 G( }Q<w> :=PolynomialRing(Q5);Q;
    2 R) @9 b6 ^8 ~! F: Z  x3 v' zEquationOrder(Q5);
    # W7 t: L3 `0 h+ ~: k3 Q/ e( P8 C: }) m( @! bM:=MaximalOrder(Q5) ;
    : I+ P) C6 w6 u7 fM;3 w2 x: ?: m% Q, A4 i" n, \9 p
    NumberField(M);
    ! v0 ]9 W6 u9 P9 X  I( o( J- 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;
      \% G% G7 j% u" L  eIsQuadratic(Q5);( |7 ~3 @  ~  O0 Q: m; |7 }* h' B8 O
    IsQuadratic(S1);
    ( G/ ]7 E" M3 O# F5 uIsQuadratic(S4);: W) P' d. D8 n6 n2 X  k. [. L
    IsQuadratic(S25);
    8 C3 H9 h0 H- O. O" s  l( l  s/ tIsQuadratic(S625888888);
    , l+ j9 z7 M! p4 Y) MFactorization(w^2+1);  
    , U7 R. q* X6 cDiscriminant(Q5) ;
    & }1 p; S! O, gFundamentalUnit(Q5) ;0 P/ l6 v+ n9 g2 c, R5 z
    FundamentalUnit(M);2 Y( s. b; P4 |' {0 I. u
    Conductor(Q5) ;9 c& i+ x7 Y( d+ X* D% ^
    ' |$ H4 Z7 `+ d- K' k) H: \
    Name(M, -1);
    9 L3 g* `+ N& @Conductor(M);
    6 }/ _3 I( ]) ?* m7 hClassGroup(Q5) ; ) ?) K# B/ F* F5 v4 k
    ClassGroup(M);9 W: e) F7 B- d
    ClassNumber(Q5) ;9 l% A2 [6 M/ E) a& @' [1 M
    ClassNumber(M) ;1 i/ J3 Y. p. Q3 j' j
    PicardGroup(M) ;9 T" D5 I2 D$ X, t
    PicardNumber(M) ;
    4 O$ L2 G0 N4 q  y, A; w. i& ^
    & r" [: M+ b" BQuadraticClassGroupTwoPart(Q5);# w  I, r0 J' o* ]7 x* i
    QuadraticClassGroupTwoPart(M);
    * l6 T3 e8 G" r2 k3 M+ f* d8 S; O" sNormEquation(Q5, -1) ;
    - a' t2 L7 S! ^/ F2 ]& `# b( R  \NormEquation(M, -1) ;$ s- X+ h: \# O7 W$ d! k/ K4 G7 O2 F
    + i% z8 R; P8 G% h
    Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    % A% I) Y  d! `, dUnivariate Polynomial Ring in w over Q5
    , a4 j  ]/ K, n8 V, i+ OEquation Order of conductor 1 in Q5. \3 F2 Z* l* O3 T
    Maximal Equation Order of Q5  V  l; H7 r/ W" T- E2 O+ m& U& L
    Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    % o4 D8 e. ~) M# Y# X  |6 b' OOrder of conductor 625888888 in Q52 \$ Z2 r9 ]) {, }: r, v
    true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field% e3 B1 r; M* V! E
    true Maximal Equation Order of Q5
    ' r, B8 f3 Q$ R% K7 B+ g% \0 Ntrue Order of conductor 1 in Q5
    7 j5 b1 ~( ?/ \true Order of conductor 1 in Q59 h+ c0 X; P3 e, V3 Y1 A5 b  G
    true Order of conductor 1 in Q5
    # }* e2 x' X' i  t! M) R. r$ d- D[
    ) V' c0 H0 ?$ A  k9 j    <w - Q5.1, 1>,0 Q7 v+ [0 n7 e4 z6 g4 k9 e0 |
        <w + Q5.1, 1>2 F% Q; _2 E4 Z+ v
    ]
    / l# E% `. ]$ F-4
    : l$ e: S+ ?+ U. ?  P7 F6 B# w7 c+ A. G
    >> FundamentalUnit(Q5) ;
    + g( ?( [9 F3 c) p3 P7 n                  ^  P. B) L4 G% J/ l! t  R0 u) r9 @3 q
    Runtime error in 'FundamentalUnit': Field must have positive discriminant; Q$ ?( B" X' P; @" J2 H

    ; Q* x7 }: g- }4 ^
    ' s9 a3 L" N6 T- J: `8 I>> FundamentalUnit(M);) V6 @: p/ {, _' e) P( H. I
                      ^
    " h: E, }0 j: S1 A1 D/ LRuntime error in 'FundamentalUnit': Field must have positive discriminant* W- G6 @; b! s: m9 i
    1 c7 f* h, t5 ~0 o
    4( R( s. A( q. {% t. q
    / I& [4 N8 p& a, l2 b
    >> Name(M, -1);) m. u8 x$ x) _) f( N0 J# r
           ^0 l" o& W/ \& C
    Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
    : \  v' o$ w; ?0 U5 S2 J% t( F; z% C# N" i/ h
    1- U+ c0 y# p  D0 |
    Abelian Group of order 1
    # X( [1 U; D8 G/ |* [( l/ vMapping from: Abelian Group of order 1 to Set of ideals of M
    6 k; f& K* z9 i( }Abelian Group of order 1
    : C4 m% \0 w. z$ R2 {# ZMapping from: Abelian Group of order 1 to Set of ideals of M
    . `4 D) l6 P" r! x; f, B18 S' O' M+ y( i4 M+ J7 Z2 P% ^$ k
    1
    ! m, K8 P8 h, v5 IAbelian Group of order 1% |* C+ M6 E) E0 j$ B' y. q
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no9 b8 ^# ~/ a( `' `. x
    inverse]
    : [% }& G% ^5 Z  o1" z: F0 j, I- N# T
    Abelian Group of order 10 Y! w& h2 K( K: b& w
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    . w2 S1 S! j2 P' |* ~-4 given by a rule [no inverse]
    ) B$ z7 B0 `( p+ A3 Q+ OAbelian Group of order 1. b5 L# z, O+ a9 U, m' p
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant6 h. r, [% L* J+ |
    -4 given by a rule [no inverse]
    , C4 S9 G4 k  ~false
    1 g7 b# d6 R3 z" Yfalse9 z% e6 b& G9 W# f
    ===============2 P7 S0 a8 Y% p# ~  T

    * k1 L* ^3 A/ |0 X5 MQ5:=QuadraticField(-3) ;* d& w) {. ^2 x$ Z3 r9 T8 j
    Q5;$ x5 i$ x( W; J/ b$ W
    5 d* i% D8 j3 f, z3 _; u2 s
    Q<w> :=PolynomialRing(Q5);Q;
    7 O/ g2 e; t2 d* X& d; ^EquationOrder(Q5);
    - h1 r/ h( a, E2 i& lM:=MaximalOrder(Q5) ;) {4 E* f( n) j3 N% I% [9 ^. c; c3 Q
    M;
    $ V4 Q6 R/ @# c9 ?NumberField(M);; I, Z: a7 K) t  r+ O4 |
    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;# V' X  j2 p+ q0 P. Z$ W7 _7 `+ {
    IsQuadratic(Q5);/ K8 f: S7 M  C9 l
    IsQuadratic(S1);
    2 h6 M5 o4 y3 GIsQuadratic(S4);
    & v7 s' u6 T3 R* u  NIsQuadratic(S25);% ]" {6 q+ f9 u6 d* L9 A6 X, c
    IsQuadratic(S625888888);
    ) ^" C# g  t- S* {8 h7 |; SFactorization(w^2+3);  % a2 e3 p" r* {
    Discriminant(Q5) ;% S/ _8 U/ Y* x* f
    FundamentalUnit(Q5) ;4 ]/ |# t4 u& h7 T5 }9 v8 z' h% a) y
    FundamentalUnit(M);
    , i0 r0 Y3 {- ~4 X; tConductor(Q5) ;
      D( S/ f8 M2 e- B. t+ o" R2 l0 A) ], X, c2 h8 B
    Name(M, -3);
    , b; [6 m- ~7 ^, A5 _$ H( hConductor(M);9 p5 u# |5 V$ ~
    ClassGroup(Q5) ;
    4 @5 N/ e/ W7 O( [ClassGroup(M);
    8 `: _4 S* o9 eClassNumber(Q5) ;2 U3 P- K1 i7 w. P- U1 R
    ClassNumber(M) ;
    . c7 [5 T! ^0 p- kPicardGroup(M) ;' o! H! g. {  ]: y
    PicardNumber(M) ;  I$ Q/ j+ e. Y' d" e" l

    ; [* z8 O- Z0 P2 v% D1 B$ kQuadraticClassGroupTwoPart(Q5);2 |3 y& n: H$ y: _- r
    QuadraticClassGroupTwoPart(M);
    ; d& }4 w3 Q7 DNormEquation(Q5, -3) ;
      \2 N( ~' b( H2 I. k5 DNormEquation(M, -3) ;. m8 s4 W0 I7 B* Q9 k
    . v$ S" a1 D( K7 k; f
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
      q' f2 B+ c  F/ A2 u7 lUnivariate Polynomial Ring in w over Q5
      O# j7 ^0 E& i  {Equation Order of conductor 2 in Q5: L. U: a5 [; `. W7 t) m, d2 o5 k
    Maximal Order of Q5  [' I4 b; J8 q# b  k- B
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    ! q, A0 ?: d7 Y; POrder of conductor 625888888 in Q5
    1 o1 m. `7 v: b" c' otrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
      w) B4 }& {- w1 x; Atrue Maximal Order of Q5) A3 ]' u9 S% l# B$ n* }% M
    true Order of conductor 16 in Q5
    * T7 F9 c: y3 Atrue Order of conductor 625 in Q5/ W% x# Y# e& k+ X5 b
    true Order of conductor 391736900121876544 in Q5; x7 c; o8 t. X, L" P
    [
    / K& h5 I' i' g    <w - Q5.1, 1>,% L( E; ]( t: ^# s! N
        <w + Q5.1, 1>
    - `# I$ n0 O. X9 U8 o: C+ F) o]8 J3 |0 f3 z4 \5 p1 Y9 e
    -3: P, g$ ~# o8 ]
    ! A9 }9 |, K7 C
    >> FundamentalUnit(Q5) ;5 _% a# A9 Q9 B( w8 w% I, y. n
                      ^% r$ F0 B0 X* v% d: [& i: {
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    8 n) m  c7 c5 M" y* @: X7 F
    2 S% V& f+ \; S9 {2 ~; e0 \) T) M9 {, s
    >> FundamentalUnit(M);
    ; G7 u( L- i0 D7 G6 x) u                  ^
      p) Z6 P( `- O$ W+ c; TRuntime error in 'FundamentalUnit': Field must have positive discriminant' }* Q+ C/ M$ `4 J4 f
    # i6 Q% h$ W  L+ Z
    37 E( g9 w' w* }9 \# y

    , z/ ~2 K0 a' X( S- b3 b/ p>> Name(M, -3);
      k. m, ~  x1 q: W" m       ^
    % |" B% c; @9 l# U6 P4 CRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
    ; T6 M: |& W1 b; n& K' x  \/ H
    7 ?% \9 o' ]% p' v8 e2 n  T1
    ( T) m% ^0 d4 x, f# sAbelian Group of order 1
    + ^" P1 H! w3 vMapping from: Abelian Group of order 1 to Set of ideals of M2 |& ?+ c& J/ J5 s/ X! x
    Abelian Group of order 1' S7 [5 _3 j; d: ]! X
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    . C, u0 D5 d6 x1
    0 M# f% s: ]3 t' Q- t* H# t) u1
    , o) [. s' x: w. L$ R* Z5 YAbelian Group of order 15 U& C; Q  ?/ i
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no: }# W$ k# {" t/ R& s5 I
    inverse]
    , ]$ D& Q) V) B0 v/ Z+ W- R1
    6 j% l0 S. W$ L7 \6 T2 I! b# G2 lAbelian Group of order 1
    8 P* ~/ d7 y$ T  b& G6 ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    3 p1 E: Q0 x( F2 s2 c3 R3 C' P! ^-3 given by a rule [no inverse]$ k+ `7 ]! l4 y: d, _
    Abelian Group of order 1
    ( ~5 U5 E. D' B. I% l2 h6 RMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    : c# s6 S$ G8 c& F  K5 m9 y-3 given by a rule [no inverse]
    3 f0 O3 h- f$ X8 P! F( p9 `5 Sfalse$ s/ P2 E9 L2 k% q/ B: `
    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
    # g. x4 @' W/ d0 a9 w# a! u/ Z/ w+ R+ z8 i1 T" @8 \
    Dirichlet character
    # u1 H( u4 v: P' @Dirichlet class number formula; X9 J4 [$ a1 r" ^
    ( o9 T) V3 T$ \  r9 H" a
    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, Y* F# r& k: p" |' W5 |* b
    1 U3 e( X2 u( k5 K5 W/ t* E; I
    -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$ f2 C3 I% |; h3 B  }
    & `; M" L6 N; k# n" `
    -3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
    5 [0 L( x% Z6 y1 Th=-6/（2*3）*Σ[1*1+（2*（-1）]=1
    9 J$ Z4 i: ^5 N( S/ t
    # X# i2 F7 t, n' N$ P-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，
    - Q/ y: J1 u+ \: f3 a4 U- Z( M/ g: l- F

      r7 D: @- w- M( d& a1 t8 d4 N
    ) F+ c. c1 e% o+ P$ j8 Q4 e7 mh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2* ^1 Y  A6 j! _# ~

    ! }# a+ U4 ~( d! I0 R4 y0 c. h  c) h0 i6 z8 J
    % Q9 G  M1 p* c
    -50时  个单位根                          N=200
    8 M; g/ X0 J1 M6 |/ o, @' N3 E* \
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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: 264)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 268)

    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: 269)

    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 " \7 v( k- B: h' g' |. M

    ; d3 I8 ~5 @6 ?; ~F := QuadraticField(NextPrime(5));
    # ~3 E% i' w) P' F6 i3 p% g7 }2 y8 I% p) c4 v
    KK := QuadraticField(7);KK;
    : v& P) t: W1 I: s, UK:=MaximalOrder(KK);
    / Y- q2 u; q9 ?( d. W/ r3 c$ DConductor(KK);1 F) e6 [) o8 l1 `: R- \, ]
    ClassGroup(KK) ;% ^% z' O/ |  I  j' r3 ^: f9 ]
    QuadraticClassGroupTwoPart(KK) ;
    5 \7 @! u( R: o$ y1 S5 sNormEquation(F, 7);' s8 r! ^" h$ I. }
    A:=K!7;A;& O: G$ y3 g- [5 N! {
    B:=K!14;B;
    . Z* M8 h  e% @1 dDiscriminant(KK)
    3 f  ?+ P- d0 Y- v1 ^- @8 W/ S3 h' d' Y  d, L4 ~) g7 r& {
    Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
    " R/ n. q0 d, \9 B! C8 z28
    # v, O2 K' u2 W& _Abelian Group of order 1' n  y4 o, X4 `9 f- [3 E% e4 N. a7 k
    Mapping from: Abelian Group of order 1 to Set of ideals of K- x! S; ]- e; H2 e8 _: s
    Abelian Group isomorphic to Z/29 H. g7 l+ W9 [( }& D) Y
    Defined on 1 generator
    8 u2 i0 g/ n/ uRelations:
    / Q* }* n7 a- D' l- o+ z    2*$.1 = 0' \" D2 c+ `4 y5 j$ a9 E
    Mapping from: Abelian Group isomorphic to Z/2
    7 H$ C1 q& r2 v5 n6 \: ADefined on 1 generator) l$ e, U( Q& f( j, O
    Relations:
    / n- V. \; V8 @" g    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
    " C0 }. t, t* \- p* Vinverse]2 Z- s# P; I# [, g: H, m: Q
    false. M" ^' b# ^2 W* M0 d! ^: {
    7
    - ]' c1 i" K7 z, g4 ~$ s14
    ' r  a+ u3 ~: `9 j( v$ _' A/ o28
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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 8 S  L" v. `1 i6 @# k

    7 S* S- E7 w; K- u 11.JPG
    & O7 x' D  ~' U7 K  ?$ |& t" o0 `  c: d
    3212.JPG
    * m  E' a1 h/ q! O1 T
    0 x# j# f3 t( n 123.JPG
    - O* G5 x2 T; ?, C8 D7 F- }$ g$ r* K
    分圆域：7 c: {% r) I- \( t7 P1 t) d
    C:=CyclotomicField(5);C;
    7 N2 |) f( x0 d) V" E/ b1 n& ACyclotomicPolynomial(5);# K! _9 K, b) o% H& r, d; ^, A5 Z
    C:=CyclotomicField(6);C;
    & K( @' f! m3 xCyclotomicPolynomial(6);
    , s$ G6 V% m' O2 A. GCC:=CyclotomicField(7);CC;
    * _4 s! m4 ~8 n9 Z3 \  [CyclotomicPolynomial(7);, A0 K  y4 Q3 y2 G2 @1 A
    MinimalField(CC!7) ;# r; g+ w5 n. v+ w8 w# u
    MinimalField(CC!8) ;# {8 Y. N6 ~4 }) K
    MinimalField(CC!9) ;1 G0 _+ b/ ^! L" ?2 G
    MinimalCyclotomicField(CC!7) ;
    ! G0 a9 k! [6 V* x0 r3 VRootOfUnity(11);RootOfUnity(111);- E: P( l" {+ s9 r7 L, ]
    Minimise(CC!123);: H: w; r# Q  _9 h5 b* ~6 U6 {4 b
    Conductor(CC) ;
    + i9 l$ @9 e8 b: I/ jCyclotomicOrder(CC) ;
    ' u' @' ]  v! u  l% O5 m9 ]* X: K2 z0 Q* J: e. d! _- Y$ @
    CyclotomicAutomorphismGroup(CC) ;* T' T( r% v$ K# S; Z

    & v8 m9 G4 g/ r" K8 KCyclotomic Field of order 5 and degree 4
    ) w3 |5 u- W2 m% r& j: X$.1^4 + $.1^3 + $.1^2 + $.1 + 15 b: w" N" z$ C% i% }
    Cyclotomic Field of order 6 and degree 2- h# i6 ~0 B" |$ H4 u3 e" t3 W
    $.1^2 - $.1 + 1
    - \# Z" w# p$ Q2 DCyclotomic Field of order 7 and degree 64 S" V$ d. M) u$ `0 }" {; H+ D9 v' P6 P
    $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    3 B8 ^8 v. Z1 \, R3 l( oRational Field
      F$ T+ {, U& L8 i/ N0 a2 cRational Field
    9 [8 g8 M& p% f9 ZRational Field$ f4 i9 N5 |! `- [; S
    Rational Field
    & E, Z. L+ K7 a9 w# _  Lzeta_11
    " s1 {3 W+ y; x9 ?  T3 c4 F- Bzeta_111
    4 q* c( x! y. T123- ^  T$ {* `' c; z* ~0 r8 e# |( u
    7: _% g8 h% m4 n. E! i* x
    75 n1 z0 c. D- w
    Permutation group acting on a set of cardinality 6& u# ]% c" ^# C7 Q6 s2 L4 }
    Order = 6 = 2 * 3
    - s2 y. c; G8 [3 ^# R4 w    (1, 2)(3, 5)(4, 6)! `, Z; |; `) A3 n& e; M9 I# h: t
        (1, 3, 6, 2, 5, 4)+ G; ]6 N8 n- P+ `5 T, _0 V
    Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of . H" [! j+ t% p. c
    CC+ A8 U9 e: w  a! K6 _8 R0 a# A4 P
    Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, 2 ^+ E! M+ v' X( S1 \4 s/ e! }
    Degree 6, Order 2 * 3 and
    ; `/ @) R1 Q- i2 T& P3 t# w% V0 RMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    / ]" ^* k' f0 v0 `. Z# T8 d# V0 I3 iCC
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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
    7 N& i9 a" s" Z* e2 S
    lilianjie 发表于 2012-1-9 20:44
      ~' I2 L; B" X6 f/ n# c分圆域：' Q. Z: F2 u7 J- Q
    C:=CyclotomicField(5);C;
    - X  d/ A) {7 K4 b4 K# j! A- `% pCyclotomicPolynomial(5);

    2 C9 k4 u; M, I& H' M; ?
    8 x' s% U) M, G$ j1 _  F分圆域：
    5 P" e) }6 g4 G! F0 Z& O" c分圆域：123
    ; u2 V4 \/ U* a9 |  O; X* Q! F2 G1 M+ |# A7 l) T! S) w" O
    R.<x> = Q[]# g( [- k- H* v# a' Q6 d7 C. _! J9 R
    F8 = factor(x^8 - 1)( C4 J% R3 m/ J# v8 y, y$ F
    F83 N. {5 S1 O5 L: g6 B

    * J3 ?$ p7 B  Q) u" O(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) , z" O& e. X! [

    # c2 ^2 `3 \) zQ<x> := QuadraticField(8);Q;! a$ l# i* j/ z
    C:=CyclotomicField(8);C;* I0 k4 F. H9 v/ v$ r5 r( U: U
    FF:=CyclotomicPolynomial(8);FF;4 j9 L4 T3 j$ }, Q& `
    0 V( P, h8 w, w# Z
    F := QuadraticField(8);5 }' {, f$ o. A/ s+ @
    F;: g1 h' @- L8 X3 ~. j# m2 Y+ P% E
    D:=Factorization(FF) ;D;
    3 l8 ^( j0 m" J( n( t3 w0 eQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field6 m' ?1 h& q* }
    Cyclotomic Field of order 8 and degree 4
    ) e8 u' C7 X7 K2 h7 x8 ?# e0 H% s- T$.1^4 + 1. x4 z  x7 k! A( x
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field! |3 d8 v* P  Z8 g& y* G4 n
    [# ^2 p% _; p/ E- y  k/ W3 N! }. K
        <$.1^4 + 1, 1>" Z7 L  s- i. k. H3 z/ r7 S
    ]
    ) G2 n6 m2 i% k
    1 e, E3 w. C% Z( D0 g  F3 P& Y5 g9 ?R.<x> = QQ[]
    ; Y6 r% h* ^- s4 u5 k) aF6 = factor(x^6 - 1)
    " v4 E& q8 F( h' ^, Y. y) h" j: M1 {F6: @- F& Z4 g/ y
    / }. D' M1 A) @- R4 E' ?( x
    (x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) : V- G3 U0 e# U
    - D0 k/ a; h: a" T' d
    Q<x> := QuadraticField(6);Q;
    3 l8 P( h8 U$ N$ G, `; ~% O/ `" e3 dC:=CyclotomicField(6);C;: o( P3 Y8 N, B8 r2 \! j
    FF:=CyclotomicPolynomial(6);FF;
    , L7 m" W% k/ [& V$ D4 l. b" k7 C$ s" ]
    F := QuadraticField(6);4 z6 g% r+ x& x# h1 n$ D, e" c
    F;
    3 K7 L2 d8 `% p$ LD:=Factorization(FF) ;D;% f" k+ K; @* c2 c+ _; j
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    2 o# L' K6 f; e/ c, _' jCyclotomic Field of order 6 and degree 26 B( I. x. @3 _2 E4 }" a
    $.1^2 - $.1 + 1  G$ x+ S2 }& V% Y  X6 X
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field" ~/ w. @" O% z$ Y% X
    [
    2 A% d6 \; M0 x    <$.1^2 - $.1 + 1, 1>( @7 F/ N6 L, l; F4 D
    ]
    ) y5 f: `& g  p" d8 Z. R4 M5 d# c% Q7 H3 B! k7 [0 Y4 D
    R.<x> = QQ[]
    ; |5 d. T4 f9 E# m1 NF5 = factor(x^10 - 1)! ]) h* h1 x; ?6 A& `5 z2 d
    F5$ G% o/ \  A4 w9 @
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
    / @9 _4 W8 F  C. _  U5 E8 m1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)3 ?+ \, y$ h# m2 t( C

    " J% U/ W6 }4 n5 i4 T: @Q<x> := QuadraticField(10);Q;
    7 ?$ O6 j# q7 u, O$ EC:=CyclotomicField(10);C;
    + E! p1 }4 G9 t# q9 E/ Y5 GFF:=CyclotomicPolynomial(10);FF;# T0 O9 {+ C8 @- ~. U3 B1 W0 ?
    ( w5 O- W3 H( [- s+ E# H
    F := QuadraticField(10);* T5 y9 W+ N+ ^
    F;0 B0 [, K2 c* ?% u+ F
    D:=Factorization(FF) ;D;
    & K  {1 U! O3 @4 T- b* \3 D1 R3 sQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    2 Y$ i" c" T( WCyclotomic Field of order 10 and degree 4
    1 [. p( E* z2 d. }$.1^4 - $.1^3 + $.1^2 - $.1 + 1
    & y- w7 i/ j1 v" r+ R: l4 OQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    ) m+ f9 ~0 u7 \# a/ h# h8 i[+ g; p& C3 z% j5 W( J3 L
        <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    * M) s0 V* ?' ?4 K, Q+ y]

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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
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    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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