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实二次域(5/50)例2

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lilianjie        

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    [LV.4]偶尔看看III

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    发表于 2012-1-4 14:05 |只看该作者 |倒序浏览
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    本帖最后由 lilianjie 于 2012-1-4 17:59 编辑
    1 y  m! e) _  ]: K" R
    7 `+ U0 u. Y2 ^+ |8 d0 P2 c7 xQ5:=QuadraticField(5) ;3 H5 a) ?! ]. P( a# ^& p
    Q5;
    , Z) `& A5 q$ ?Q<w> :=PolynomialRing(Q5);Q;
    ( |% a7 L( v+ G7 ?: T+ y% |5 F
    8 N! J+ c- d0 l& QEquationOrder(Q5);
    1 @; {8 Z- w8 [3 VM:=MaximalOrder(Q5) ;0 P4 T; D  r9 {  J3 U+ e2 @
    M;
    2 ^1 D  o/ g+ @. m0 V* J6 xNumberField(M);  C  z1 {& t! P& c# V6 ^% s
    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;
    , L5 o. b% ^2 [) {0 q% qIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);
    ! r* w& R& E: L# F2 [  h: i9 X2 u& }Factorization(w^2-3);; S0 h- |) U3 s* b: p
    Discriminant(Q5) ;
    $ J& ~: L' r) B2 m/ ]. j7 F, pFundamentalUnit(Q5) ;* a# g  ]) V9 e' w2 u. M: H& x
    FundamentalUnit(M);5 b( ^' f) h4 F- E0 R! x$ ]. p) i
    Conductor(Q5) ;1 ~1 m+ j6 N" O1 G# |: J4 s
    Name(Q5, 1);0 m9 k+ B8 |% p% A
    Name(M, 1);
    8 G: V2 ]& Q& d# SConductor(M);7 P9 ?& v: A' j6 A( |! q
    ClassGroup(Q5) ;( e% f* B5 A9 @/ e) H4 F
    ClassGroup(M);9 l/ z5 K! c' j6 p* w+ x# z
    ClassNumber(Q5) ;. {# ?5 k4 I& _5 b0 e4 Z1 Z
    ClassNumber(M) ;9 s: \! }/ y2 u

    8 G9 b6 x2 r( \9 @+ g. ePicardGroup(M) ;
    % u" J; j" |# ~& P1 FPicardNumber(M) ;
      J  P( ]6 F9 y) s/ c) b- N0 A- X0 y, y1 p
    ; m5 Q5 W6 i  I% ?0 H4 l
    QuadraticClassGroupTwoPart(Q5);
    - g0 {* H, S2 zQuadraticClassGroupTwoPart(M);1 K6 @9 }  F. y9 i  F  g

    : x. r5 w4 e! H) r5 @3 ^" P% J: C
    " A& [+ W1 X; Z* W  z$ wNormEquation(Q5, 5) ;
    9 Y$ D, F/ h5 _' q2 BNormEquation(M, 5) ;! |: c1 N7 n' Z& l+ ^0 r2 I! ]
    3 i0 ]5 t  o  Q, S5 `: N
    9 I) z' x0 |, a0 m* m
    Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field, b# R7 r2 |* O! F
    Univariate Polynomial Ring in w over Q5
    1 T' S+ N5 W) |4 B2 N$ ~$ b; [Equation Order of conductor 2 in Q5( K5 \2 H' z' o9 w& E' w+ C" M
    Maximal Order of Q5: S/ P* ]! W0 O- A( W. ^
    Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
    / O& T/ u+ R* Y2 JOrder of conductor 625888888 in Q5$ u0 S1 e. G( N/ R( n2 T
    true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field# O4 w  H3 s* Q% G4 ~$ n
    true Maximal Order of Q5
    % i# p9 v$ f) c9 w$ Ktrue Order of conductor 16 in Q5
    ( t' }, f5 |' P4 S/ @9 ltrue Order of conductor 625 in Q5/ n. X, g' V. }3 X6 J
    true Order of conductor 391736900121876544 in Q53 k: q  S2 `. M
    [: `5 c. O# O. Q6 J
        <w^2 - 3, 1>5 j+ _  _% P6 a6 n
    ]" U. A+ [" h6 i+ Q2 Q0 k
    5
    9 o% @: ?, Y2 Z$ k! R5 R; e( N/ w# J1/2*(-Q5.1 + 1). r- p& y7 p: [% ?6 v* [
    -$.2 + 14 B; i# Y3 W: A* i% x+ V6 |) i
    5$ K4 u$ K4 ]3 H, @# r" I
    Q5.1- C5 {, O, d' t9 _: }
    $.2
    3 D9 w- o! v. D6 I- P1
    ! k/ N& q# x+ b0 mAbelian Group of order 15 D, T- }& E& k9 M
    Mapping from: Abelian Group of order 1 to Set of ideals of M, m8 m  r$ c$ s3 O# D+ ?. k  k
    Abelian Group of order 15 w/ b, ?$ f% g, H* v4 E" U4 X
    Mapping from: Abelian Group of order 1 to Set of ideals of M. b$ c+ B( M* ]' k
    1
    1 {( G9 ~4 L: k" x" n4 g  j9 R1. C0 ?+ v* O: h$ u
    Abelian Group of order 12 b8 j3 M- ?8 p
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no4 `: w+ j; ?% u0 c" D
    inverse]
    * k4 K3 W% H, g+ N4 `2 ^1  B) c' @# P9 o3 [, C2 I- A
    Abelian Group of order 1
    ( t. q: J+ {0 v; E4 Y4 BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant; Q, W: x& k9 n6 U* w
    5 given by a rule [no inverse]
    7 U( z: r5 C6 W5 R/ J$ J+ T) O; yAbelian Group of order 1
    * Y& T; M& _8 L. O) iMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    5 `6 [8 V+ I( @; }$ A' O5 given by a rule [no inverse]
    : Q4 L5 m0 W! o; G: x) Ptrue [ 1/2*(Q5.1 + 5) ]1 B7 U6 o* t4 }! j# A- b
    true [ -2*$.2 + 1 ]
    - L1 H8 _; O  F7 Q9 S' r" L/ N
    % U# V7 p/ k& G6 f5 e
    6 v3 X5 N+ Z7 d8 P4 Y1 |
    1 W: c  p- }' c) A/ O  h. I8 ]% ]1 Q" E  u7 q4 {

    4 I; L% w+ C7 k, g
    9 w; C  J6 n5 E. Y) U' N) e- E* s; _. a* n1 {/ N& f$ U' y; r4 k9 p) X

    8 Q$ S' C( W& B! A% d2 I2 X) X. E8 |& [

    . \1 a9 z' O# e' \# b% q( f) N5 o
    2 ~4 i( S# j% h7 h' o8 Y==============
    - L5 c9 @  h. C8 O0 ?7 I
    # d" S5 ?6 Y! g9 T$ |Q5:=QuadraticField(50) ;# T1 [4 v+ C, m0 H# v
    Q5;
    2 I  q# {; y# n" p5 z6 u) U+ J
    ; T' V+ t- j2 D" J! _" [Q<w> :=PolynomialRing(Q5);Q;
    6 D. U3 w9 @2 w- Z% ]( ZEquationOrder(Q5);
      M# T2 |, g7 F, @M:=MaximalOrder(Q5) ;
    ; J0 T  H9 h% ?* MM;6 W# j2 C" C% k6 X. |; L+ j0 E- T
    NumberField(M);
    : M! Q7 k2 C1 K; d& j& sS1:=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 c7 p. ^% E$ q2 j: o: \7 q& \, cIsQuadratic(Q5);+ u  ~% k) T' [7 d* E
    IsQuadratic(S1);' I0 T3 t0 O& ?$ n/ O( ]
    IsQuadratic(S4);2 o3 H" E1 `+ Q
    IsQuadratic(S25);8 s: M. R& t, Z" X4 X" O0 u3 x
    IsQuadratic(S625888888);
    & x4 a5 n8 y4 ], h4 G$ WFactorization(w^2-50);  
    ) G7 ?' Y( H8 Q5 t: N( fDiscriminant(Q5) ;/ g- ]2 w/ s' |& P: d: z1 d  S9 @& B
    FundamentalUnit(Q5) ;
    7 |; T3 A( U1 Y' C( sFundamentalUnit(M);
    ; p1 X: l; _+ N1 c' c, lConductor(Q5) ;* A+ _; A+ ~; m
    4 i  X3 X0 {) T" N7 |9 k" J2 D7 J
    Name(M, 50);
    - @5 h1 L: ]* a$ K# h. E* BConductor(M);" ^& j" x1 v! t3 t4 U( p
    ClassGroup(Q5) ; 8 m: \  x  {6 [, r
    ClassGroup(M);/ S' C1 ]7 ~* R/ H
    ClassNumber(Q5) ;
    / B* ?2 u5 j, n, n. d: VClassNumber(M) ;+ ?. ~$ H2 |! q% W4 D+ h
    PicardGroup(M) ;
    8 R4 {! o! r5 zPicardNumber(M) ;7 j' x, `3 |+ t5 g( e

    + I  w. }6 F4 R4 X' `; kQuadraticClassGroupTwoPart(Q5);
    & O5 P: y6 c3 B1 H2 H. [QuadraticClassGroupTwoPart(M);
    . s" g1 N  I4 b" I' RNormEquation(Q5, 50) ;5 r3 O2 e% l$ [
    NormEquation(M, 50) ;/ r0 P8 a# ~. N9 D

    ) e+ U1 c4 i, ]Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field* y" w/ D/ S. C  Y
    Univariate Polynomial Ring in w over Q5- Y2 v/ \3 D5 R
    Equation Order of conductor 1 in Q5
    3 y9 x  F( y8 wMaximal Equation Order of Q5
    : E5 ^6 K! f! Q  c% o# m( K$ |9 C% CQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field( U' N3 |+ A& B
    Order of conductor 625888888 in Q5
    6 D2 _# G1 Y7 _, {- @true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field3 [# o$ h5 G. i  V
    true Maximal Equation Order of Q5
    / B' [) x- A* Jtrue Order of conductor 1 in Q5
    + c, p5 m+ F5 k/ ltrue Order of conductor 1 in Q5) h6 w+ k' @% \9 F6 y
    true Order of conductor 1 in Q5& Y/ b/ @0 p% l8 D  w$ B8 w
    [
    0 q/ Q: @0 T& h1 ]4 m* Y    <w - 5*Q5.1, 1>,) b' Y! L- G7 X+ g* ^# r" D. g- C$ q
        <w + 5*Q5.1, 1>' a) D( j- V, f3 ?. f* v! W! B$ k
    ]
    ( j4 B& V* O4 J( J6 L8
    ; b6 H. b2 C2 }Q5.1 + 1
    % ^& I  w# g' m; M6 q. K$ ^2 Z$.2 + 1
    " M( ?* D& |  M# _8
    - T2 L  f* C# }) {& H' k: C3 J6 P& b- B& B, z4 p+ ~* H
    >> Name(M, 50);
    . G6 k& f6 {. A# S+ v, i5 C- F" S       ^
    , K, `8 A4 B7 o# a! h5 r, nRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]0 n' f$ D+ @' n* J7 m/ U( |2 n

    5 d7 c9 J$ u" i- Z$ V9 l7 `: e1
    " `/ U- ^0 Q; r  i0 C( L3 D4 u* dAbelian Group of order 1
    ( z' d8 j  e  t: L8 p6 ?$ a% W2 @Mapping from: Abelian Group of order 1 to Set of ideals of M
    / H3 V! J$ l5 m5 c5 U+ O. o1 jAbelian Group of order 1. n  r7 A. E7 W4 G& H" v  f0 `3 b2 x
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    % p- C3 x3 D, J7 G0 H: _8 H1 \: T1
    , x& S6 [5 Y* k5 ^* n1
    % F& h: ]$ ?- P: H% r* uAbelian Group of order 1
    - I3 A. v- [; s$ t0 wMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    ( `1 ~4 t, \: H* \inverse]
      C+ o. j) T+ k9 x& d& ~# D* _8 `1
    " t! y  V' ^# _) G  w) TAbelian Group of order 1
    5 R5 G5 J) `5 ^% ~% ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant$ v! ~0 q* w+ N
    8 given by a rule [no inverse]
    - ~. Z. @: W) P( ?" Z2 n/ Q& ?Abelian Group of order 19 `7 v+ s; }2 i' d1 A4 q: C
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant' {2 y7 s0 l* l% K
    8 given by a rule [no inverse]
    . U. S- B( ]  strue [ 5*Q5.1 + 10 ]
    $ r- X1 P7 l7 N  Xtrue [ -5*$.2 ]
    zan
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    lilianjie        

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    [LV.4]偶尔看看III

    二次域上的分歧理论

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    lilianjie        

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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-5 12:42 编辑 ) h4 r) G8 [1 b+ l! n
    : q# L7 k4 [6 h1 L" h
    基本单位计算fundamentalunit :$ m7 v. D+ Y" O, s2 v
    5 mod4 =1                                              50 mod 4=24 v! U- }! K) `/ l

    : u0 q& J7 F- o- P$ T& D: c; \) w x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
      ^0 W4 G: N, D* E8 p' m# B- d" W x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.: S7 }: v. f' r' [
    $ o* ~- d1 ]% `5 ^
    7 b. k" |# P% s. R
    最小整解(±2,±1)                              最小整解(±7,±1)
    6 H0 l; [8 |8 `9 p0 n8 l5 I! O+ ^6 I                                                             ±7 MOD2=1
    ! V& |7 y  H4 ~& B4 i' I5 ?9 q) R
    ! s: o! Q" w6 O" ^' n( k; s两个基本单位:

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    [LV.3]偶尔看看II

    lilianjie 发表于 2012-1-4 18:31
    1 T# P( O5 m/ b基本单位fundamentalunit :% U1 G! G' L' W! {' \
    5 mod4 =1                              50 mod 4=2
    9 A( n: K% |/ Z1 C# e0 D- P/ S. K3 K
    基本单位fundamentalunit

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    [LV.3]偶尔看看II

    本帖最后由 lilianjie1 于 2012-1-4 19:16 编辑 % P- V: G! n1 }7 M9 z
    % j/ y* }6 T8 V% ~+ y1 Y
    判别式计算Discriminant# p4 {( w- _" m' I& U

    3 Z5 f" d+ B% N+ ]5MOD 4=1
    ! H! n+ C: o& D, w2 R
    5 L$ n& S' W! b! N3 K(1+1)/2=1          (1-1)/2=0" z' f) p$ p% o$ x

    ! O; X$ u/ O+ n  r* L$ h  sD=53 P, G' y% A& N5 l& g) \" @4 j, o
    4 K2 c4 q7 J: d
    5 H9 G5 {9 ^: }2 `& _8 R$ s$ Q, U
    50MOD 4=2
    ! V# x, V1 j' u7 U7 MD=2*4=8

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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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    [LV.4]偶尔看看III

    lilianjie 发表于 2012-1-9 20:44 ' ]& f# }5 I3 g

    : e) [3 l7 w, [$ g+ B( j. n4 G* `分圆多项式总是原多项式因子:
    6 e0 r' w+ @9 AC:=CyclotomicField(5);C;
    - [* j9 M6 J( m; `& `/ H( OCyclotomicPolynomial(5);
    . \: R# o9 ?! G. d1 y

    4 B- Q7 g0 d5 W; F& Y分圆域:
    1 C$ @) M1 T7 K; T) `4 d分圆域:123
    % O0 U$ E! Z$ @8 _! P% v; @/ j$ x4 k  @. @% Q7 c! P' W) ^* q
    R.<x> = Q[]
    ; w) D, f1 h1 p# v/ JF8 = factor(x^8 - 1)/ W. ?3 v5 E1 O! h% j% j5 B4 }. M
    F8
    : ~2 ~- i* `$ }* p* w3 }. m! Y0 t( Z9 [9 R( r' r
    (x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) ( N/ ^- j4 x/ w! ~8 @5 P3 P& `4 t5 U

    # I# O: x1 T' pQ<x> := QuadraticField(8);Q;
    ' l/ h$ @- E8 g9 A( S+ KC:=CyclotomicField(8);C;5 F, `6 j( D, r! r3 z
    FF:=CyclotomicPolynomial(8);FF;
    8 g) V. s: Z5 e0 s. }, m/ t6 B: T" ?! p' f
    F := QuadraticField(8);
    / t; k' e0 O4 AF;
    ' H4 {  B2 f2 l5 R; r6 J9 VD:=Factorization(FF) ;D;
    $ @3 C. D1 S+ n! L9 {& r& cQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    9 b* @; q7 b* d/ {! XCyclotomic Field of order 8 and degree 4
    8 h8 P+ x; K$ |3 n9 a9 s$.1^4 + 1" [4 L# `* p) h5 u# Q
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    ; t1 X( V5 ?+ G( C9 w[
    , S2 F5 K2 @4 A1 i; U; C    <$.1^4 + 1, 1>
    2 ~( J) R0 w; j1 C( N& d]( X, U+ I1 `' |; x" G

    8 y/ @" Q- ~& F3 KR.<x> = QQ[]( ]$ g+ C2 w+ F1 U8 n
    F6 = factor(x^6 - 1)
    3 u& F- o2 B# ]" Y# ^F64 f2 d% r  ]- v  Z4 i3 {

    , H9 M5 k7 |! r(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) . A9 ^6 H( S- ~
    , s. z& R1 e2 ~! N1 ?
    Q<x> := QuadraticField(6);Q;
    ' K' Z& Q1 F0 WC:=CyclotomicField(6);C;# @6 a% L: Z: E3 C  p' Q/ H; r* Z
    FF:=CyclotomicPolynomial(6);FF;
    1 v+ ], _3 P( D8 O- K: y
    2 x$ }% N! K; Z/ Q/ IF := QuadraticField(6);
    + l4 w/ P5 R6 A$ K3 s8 EF;/ h+ t7 u9 y2 U( g% B# Z
    D:=Factorization(FF) ;D;
      F. E" z( R8 Y1 ?% T! FQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    % P9 W  f3 t7 o4 g- W6 iCyclotomic Field of order 6 and degree 2
    ! t* P7 [# W+ T7 h# N$.1^2 - $.1 + 11 ~& j2 L$ i  a# Q- |+ t8 e; n
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    8 x0 J  q' j" f9 Z( ^) J1 a# ]2 H[
    & y- L& u' j# Z, k5 M    <$.1^2 - $.1 + 1, 1>
    8 d* d" p4 O, f* D% q1 x  o" h* y]& H; S  k% [7 `3 g' x) E

    , G4 A, X8 v4 w9 t/ i- SR.<x> = QQ[]0 H& Y' k# e% F% c0 L
    F5 = factor(x^10 - 1)
    & T1 h& ]* \" V  y5 r) `* C, jF5$ k6 ]. p" w7 R( f& n0 n2 h) y* Q
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +9 V- U; r: Y8 U% W0 o
    1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)8 p6 w+ c) x7 H( E
    ; y# Z1 z8 Z9 w/ q* Y! s- n
    Q<x> := QuadraticField(10);Q;- X: r* X+ R! _/ Q$ v; a& v/ x1 l) v
    C:=CyclotomicField(10);C;5 Q9 n0 q: m0 P5 L7 [# C
    FF:=CyclotomicPolynomial(10);FF;7 ?, M( X! Y0 b# {

    ; c, o; I( B9 w7 n; cF := QuadraticField(10);# T8 G' p: X! S# P" D$ E
    F;: |; n5 W" B5 `/ o8 w# @& D" _* V# V
    D:=Factorization(FF) ;D;
    % R& Z/ A0 \6 IQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field# U6 Z, [) L2 n9 c; C
    Cyclotomic Field of order 10 and degree 4
    4 \$ j9 Z! A( d& y( f$.1^4 - $.1^3 + $.1^2 - $.1 + 1
    ' n9 r4 [8 L4 a0 P' }. x1 N# w  c' VQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    3 ]; h+ q: {" f7 u+ R+ }[
    " I7 U) W; j7 F0 Q    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    # D. X. N; m2 x: k2 D# x8 _1 Y]
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