虚二次域例两（-5/50）

lilianjie

3 P" B7 z" U' \9 `
Q5:=QuadraticField(-5) ;
Q5;

" K0 e5 w, \! B' u) {Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
2 u) r  x/ n* a7 gM:=MaximalOrder(Q5) ;
) k2 I1 v+ L2 g0 u8 zM;
3 u2 [% @% L7 W& TNumberField(M);
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
+ ~7 m; ~6 i! @5 e: GIsQuadratic(Q5);
+ ]  j) x* L' SIsQuadratic(S1);
IsQuadratic(S4);
, f8 U$ x4 \8 z; |8 w' Q! ?# n8 rIsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
7 F" ^6 X! L7 t3 \4 _1 mConductor(Q5) ;

Name(M, -5);
% z9 C5 C! O; ]4 kConductor(M);
8 b# }6 X+ V% Q* TClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
' T5 F! n+ _% ^8 M" @ClassNumber(M) ;
. D8 b, R2 t+ p* t: \3 d" X# `PicardGroup(M) ;
PicardNumber(M) ;
9 }  {$ b( g( x$ k
  G  g5 p2 I) h# rQuadraticClassGroupTwoPart(Q5);
. u. o: O/ R) BQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
* ^. n1 s: Y8 e1 y6 sQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
  w$ y! N) v% N( F7 C- o3 VUnivariate Polynomial Ring in w over Q5
: g5 ]2 z% _$ n1 p+ ?, g8 kEquation Order of conductor 1 in Q5
; E9 s2 i$ o2 w8 @. Y* V: ZMaximal Equation Order of Q5
7 a; `+ W$ z  q: A% ]1 |Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
  g# J, S- k9 z: E; J( `true Order of conductor 1 in Q5
, u/ X/ q+ u4 }! r/ R$ Ltrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
, D" b+ h5 y# m% l9 t- I[
# l4 Y! u7 Y  d9 J, Q    <w - Q5.1, 1>,
    <w + Q5.1, 1>
4 g9 b' C- X* N- a) ^]
" x+ d# R7 S& A- V8 m-20

>> FundamentalUnit(Q5) ;
; [& {' S' C" C7 b" X                  ^
! F  H7 l; B: @3 vRuntime error in 'FundamentalUnit': Field must have positive discriminant
! O, W0 ], A; B% W0 D8 A. g8 L
' L6 r4 R  s# z+ R
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
! K6 s& r; N! O. K3 S
) j. d5 s" [2 }- l6 f8 q20
% H4 j0 O) O7 {+ U/ }! Y( K
4 p1 [. S1 U" i9 W; M( L0 g8 i. Z8 V, U>> Name(M, -5);
       ^
- u, f# R8 d' C9 f* l/ P% s( iRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

, L; f3 d& Q( N+ E0 s6 G4 L3 E( f1
! G" o- B; R7 Z6 ]! XAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
& n0 t: W. s) |& C* B. ?) u2 oAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
( K3 Z; w! C3 T" r  G) \3 Y* y    2*$.1 = 0
* X+ j# F. M7 T$ Y2 S; {  J, @Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
" I$ F# ^/ I; g7 Y5 v& O7 b) D. n3 NRelations:
2 I6 S) Y8 J4 G5 F/ T* J+ Y* d  K    2*$.1 = 0 to Set of ideals of M
2
) m. @7 V( J, W+ `, B2
Abelian Group isomorphic to Z/2
1 ^( z7 [$ B: [Defined on 1 generator
7 f, i, M( R: FRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
7 h; {3 H! W; }7 r# ~; z8 }" lDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
% f/ P( m% C' [2 ^( a% |7 ^8 \1 ]& fAbelian Group isomorphic to Z/2
' ~/ k. S8 ^) A1 X. JDefined on 1 generator
Relations:
    2*$.1 = 0
4 I" P, s+ v* n5 KMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
' W# z& f. c& u- Q& N/ FRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
9 N$ N' K" M, Z, L1 N: qinverse]
  @7 _; ^  h2 }# q  ]4 l$ YAbelian Group isomorphic to Z/2
2 \3 w) S% [" W6 d# g' @Defined on 1 generator
Relations:
8 |# z1 V! y6 W& @  a' J    2*$.1 = 0
& g& `* ?) A2 x  m$ u* a6 s8 q; D/ xMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
  E, m/ Q2 f* x; h! }Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
9 Q  o3 ?* O) X6 s) W, N6 W7 v4 kinverse]
( M, {( R  O, z5 Cfalse
, l" I3 B, U8 s) Gfalse
==============
6 I6 `; ^7 [/ ?" w
# Z, Q; _' Q, L$ ^% ^
Q5:=QuadraticField(-50) ;
Q5;

# ]* C% D; Z7 ^8 W; K: Y, O9 rQ<w> :=PolynomialRing(Q5);Q;
) s6 I+ _" d) U# n& {EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(M);
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
* p+ K9 J: H7 ?: U- H1 {  tIsQuadratic(Q5);
% z2 e1 w; `: S4 S1 gIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
, P" s2 ]7 J" o  l- wDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
: |3 f' O) X5 G/ `. [( [' h6 fFundamentalUnit(M);
7 H( C" k* k" T/ y- G1 R: x( H4 pConductor(Q5) ;

; C7 z3 X- x/ z/ LName(M, -50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
$ C& x( {* E4 F* T1 `
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
* [  L6 S2 J( z0 u( B* O8 W+ WNormEquation(Q5, -50) ;
NormEquation(M, -50) ;
& Y& z9 D, N( N9 U9 T- M# u- S
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
) W9 U6 n0 O( ^7 K2 iMaximal Equation Order of Q5
. R$ [( U# `. I6 O- BQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
( y) l* W* J2 n* {' B3 ROrder of conductor 625888888 in Q5
0 l  ^2 _6 Y  {- v9 wtrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
4 T6 x* d+ |) e  ~true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
9 z6 L& n1 {0 o9 S7 q  a& H$ [true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
6 A+ w2 _! U0 _3 n2 D[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
8 g% [, D! @( b5 d# l8 K]
-8

6 t, V/ B6 |7 P>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

# L2 A$ Q2 L" M" V
( s0 u5 ?( |! f5 Z. L/ K; y1 z: o; Y>> FundamentalUnit(M);
                  ^
# I8 g$ @) }+ W5 E8 z6 ^Runtime error in 'FundamentalUnit': Field must have positive discriminant

8
* K5 z% y- e1 M8 @
>> Name(M, -50);
       ^
3 K2 u2 F8 R7 T1 i3 jRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
3 S# i4 |: q% }: I( m
% a7 M6 a1 l, K+ t; x% D' U' F- ?1
  z0 `9 ]3 p9 N  Y  L$ F* X  y3 SAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
/ J, r8 S  r2 X) A: o) D+ C! ?Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
. y: t& k" p" k. u+ Q% QAbelian Group of order 1
, `7 X. Q& l; OMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
; S9 J+ s# \: X; f7 W8 Oinverse]
1
1 R+ P7 Y( t- P7 P' _8 f; U: U5 zAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
: z* K1 f% r/ |5 c. p& V% S-8 given by a rule [no inverse]
Abelian Group of order 1
6 k9 ^+ a7 {/ H+ n0 AMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- F9 v' p9 p& r2 \) _  f8 Z-8 given by a rule [no inverse]
false
& p1 _( N- ]- G3 d/ Y* l+ efalse
5 E& @+ y0 U# V2 [. q+ }3 w. s

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;
7 v1 V+ u# U! v) V, e
Q<w> :=PolynomialRing(Q5);Q;
3 [# C1 p1 b+ y: \6 ~EquationOrder(Q5);
* _! k/ @3 ?. k- E4 |* G8 oM:=MaximalOrder(Q5) ;
M;
! o, g" d$ R4 y: ZNumberField(M);
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
1 J, p9 f6 h  i% l8 xIsQuadratic(Q5);
+ L1 E0 d) x8 r) s- S- J4 CIsQuadratic(S1);
: g7 w- w+ _% t- z# w% x/ }' F" {$ }% \IsQuadratic(S4);
, |* u) b, ?; J2 [8 A8 fIsQuadratic(S25);
# N  b1 I( s. C+ E2 EIsQuadratic(S625888888);
Factorization(w^2+1);  
/ h4 ]. {; ]) T0 e/ f0 c  P/ EDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
* v8 o. `/ r+ S: }Conductor(Q5) ;
. P( I' |2 p' N: v9 G& g0 o
; q3 c# e% ?9 X5 Z8 GName(M, -1);
6 \# W9 g$ E% R9 w. YConductor(M);
# N3 e) H6 x% X7 Z  P9 `# KClassGroup(Q5) ;
) b& M1 ~& g: {4 f) B* }4 y- AClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
1 ~9 D' M% d* k6 |0 I6 A, MPicardGroup(M) ;
) K1 H' \8 [) X' ]) R  k) P6 u; B  _PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
  g5 l- J* f1 [QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
$ R% Q$ ?* X  J6 xEquation Order of conductor 1 in Q5
+ ?% }2 b! P0 g" P9 Z. hMaximal Equation Order of Q5
- h, m' `9 T  [Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
8 D. h& H, Z, ktrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
1 L1 U/ u4 H# _* htrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
9 S/ i) M* K9 p+ U: K0 c* q1 |true Order of conductor 1 in Q5
* N* M! U; O" r& D3 Ztrue Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
- u# a& g; B9 p8 O2 g    <w + Q5.1, 1>
]
  i5 K% r. z2 X9 S7 d2 k-4
7 E+ `. m- u, J6 a, {- u! ?6 w
>> FundamentalUnit(Q5) ;
; C& f# H/ f2 m" b% r$ C' P* \                  ^
5 t0 C9 d  P1 ]; dRuntime error in 'FundamentalUnit': Field must have positive discriminant
- A+ u" A5 t" ?5 h& i. I
, i" G6 d# Z3 U7 z
; p/ v2 ~5 x+ A( l- O% y1 t>> FundamentalUnit(M);
0 G0 {# A5 U" [% R% j                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

( w6 w- `: |3 T* L6 j; P9 k9 ~) C4
0 z4 U" i5 @2 s7 i, b" O5 H
>> Name(M, -1);
+ r- \* A3 N; q3 L       ^
- A& q1 w8 R5 ^% P$ k# e2 v, j% n- sRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

8 m2 @. X, b& w" F3 {  r& O1 b1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
4 a  L$ V+ z5 m) y% SAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; G2 d. h& m- I-4 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
false
===============

/ J% ^- d: N( W5 `' rQ5:=QuadraticField(-3) ;
; |2 q( V/ N2 A6 `) xQ5;

2 j6 p5 z) i$ y0 c( y5 L" m! Z% M0 RQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
4 }- Q- Y- \5 P) M5 aM:=MaximalOrder(Q5) ;
M;
4 e  B& C& u$ v9 u4 ~+ o% tNumberField(M);
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;
" A+ Q. a+ n1 Q" x2 KIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
% p# ]  i; K% D7 KFactorization(w^2+3);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
2 B1 F2 U+ Z% ~! `; R: X
Name(M, -3);
4 E8 a# e9 b& A( w3 _7 _Conductor(M);
! w( D$ V! ?3 ~' oClassGroup(Q5) ;
ClassGroup(M);
3 r; a& z+ G3 l( LClassNumber(Q5) ;
ClassNumber(M) ;
; N8 [# k" ^9 x7 NPicardGroup(M) ;
PicardNumber(M) ;
& K$ `; [# v  U0 i, J- n7 V
/ ]% R, k; R$ p) e" P- ~QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
2 M: o& z  B6 oNormEquation(Q5, -3) ;
NormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
2 g9 `( F/ Y4 D  WUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
% E( m1 m$ y2 Y! }; U) lMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
' d, b$ z! l2 E0 s3 u' TOrder of conductor 625888888 in Q5
( G+ W. o% r. ^; w9 etrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
! c3 P; W9 l; I' @, T* R/ E3 N4 Ktrue Order of conductor 16 in Q5
2 `7 }% b1 H, t& M: I8 _true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
7 t; v; W* v) F9 m: B( C[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
' c' }* ^/ d: \]
-3
9 `+ d  h8 h! e/ P6 B* l9 p) z
/ ~" ^( i' k) l0 j9 C, k>> FundamentalUnit(Q5) ;
% S8 }. k# w1 z6 f: g                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
* a8 t  O# a/ T0 N' ?5 u# F  [

>> FundamentalUnit(M);
                  ^
3 {, v8 c9 U6 kRuntime error in 'FundamentalUnit': Field must have positive discriminant

6 P0 u3 v, q# ~# `$ e3
/ g% X. b, V  n" `8 l4 _7 J
>> Name(M, -3);
       ^
3 V0 N* j5 r) q: v! e# @Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

% h7 f3 p9 A. t) G1
Abelian Group of order 1
$ _7 x8 @% V; y" M2 `Mapping from: Abelian Group of order 1 to Set of ideals of M
- z4 t0 ]0 o; w9 Z- J1 a' |9 m( JAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
1 k. ^* V9 \2 a% {8 MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
8 }# H; m" J8 n5 L/ b8 }  Zinverse]
: i0 r( x# N/ M( F1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
# C* I& `6 q$ v: H-3 given by a rule [no inverse]
Abelian Group of order 1
, t1 t  j' W. k9 h3 B% BMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! V% v1 h1 Q7 p2 j1 }( ^-3 given by a rule [no inverse]
" F, N3 D* }1 w( i- ufalse
, `7 k8 g4 {% pfalse

孤寂冷逍遥

lilianjie

. ~7 ]+ n& z- R8 R+ n
Dirichlet character
Dirichlet class number formula
- ^9 @+ Q  ?: ^4 Z. V' a" {
) n4 ^3 o- f# J+ y; U1 W! }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
% h9 ^* F+ p: N
* _6 f6 [6 V6 k- x* U( }6 i& F% g6 p-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
% W, l: j* e$ j. Q8 p( ]
( z1 F/ O5 z! m5 a+ w. O-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
h=-6/（2*3）*Σ[1*1+（2*（-1）]=1
' D0 S! B% U# Z
* A3 M6 m9 ^" Q" r1 n; _-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，

4 O1 ^/ U, ~0 u
; {9 }& [7 _" N4 _
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
4 b* B- L- X8 M/ ~

, f0 v4 f0 X; j. a
$ A6 _; V- J0 q7 W1 ]4 P-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

- _! I+ _; H% b* _9 O
F := QuadraticField(NextPrime(5));
8 M1 N3 P6 @0 M
* K" l  r$ M; p7 {' s' K8 S2 mKK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
8 }: r' W/ C- q+ L& U, i9 pClassGroup(KK) ;
3 y: F/ r/ X0 S' K* o% I& IQuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
1 d+ `: U) ?7 d# y' t# ?B:=K!14;B;
) D( O2 c: I/ M0 [6 `) ]& CDiscriminant(KK)

' n  D9 `) p: N  W; B" O1 oQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
  p/ I& }$ Y" q    2*$.1 = 0
3 O9 O, C2 [8 `7 o, pMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
1 E$ f/ G3 j. @4 y    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
14
4 C; G* p7 H% H& [% F+ a; U7 ^3 m28

lilianjie

0 y! R7 E8 B  i; S0 [% m
( u5 x6 q3 \2 I* ?' s2 }, F

8 G  `8 m; H; ?" s7 w, d

( Y  Y8 _7 ~& z. Y7 A2 A; L
分圆域：
C:=CyclotomicField(5);C;
$ O7 `, d# |5 Q+ O/ v. HCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
5 C" O* C  k6 C2 m8 Y" q$ GCyclotomicPolynomial(6);
4 R  }  O, L" M, S+ F+ }9 W- WCC:=CyclotomicField(7);CC;
" w' x2 k6 f- U* d. x2 @CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
0 {9 S2 K1 Z. Q) W. B/ CConductor(CC) ;
CyclotomicOrder(CC) ;
  u( `  n: l3 z  g  R7 }3 Q0 L$ ]& r
CyclotomicAutomorphismGroup(CC) ;
+ [9 @. a1 q5 C2 G( S8 |4 v
Cyclotomic Field of order 5 and degree 4
+ y7 {, Y% n3 n* ~( y3 f; Z! @$.1^4 + $.1^3 + $.1^2 + $.1 + 1
  A3 A) f! G- C) W& oCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
4 j$ k. S1 x  C2 W& ^( r; _$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
Rational Field
+ K1 p) r: L! J, @% K. D4 m8 [zeta_11
zeta_111
123
7
7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
0 [# I6 Q, z  I    (1, 2)(3, 5)(4, 6)
9 v7 Q; V7 O5 z; G( T    (1, 3, 6, 2, 5, 4)
" B/ ?- o8 F* M# uMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
! b8 K1 L" C  F/ R6 Z( Q. O" wCC
) m6 `8 L+ {; q; E$ TComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
, r3 _: h  V5 B9 }+ V! w* ~7 oMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

1 e( {* X( ^) H* W5 b
# k8 J2 S) Z, e* z分圆域：
分圆域：123
' A6 c/ I0 O9 C- t
% N# |% s% Z; bR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
2 d. J5 K" U0 M; O  N) p$ n: [
# F$ \0 `9 R% M) x(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)

Q<x> := QuadraticField(8);Q;
. l  P: @- e4 f  W+ \- NC:=CyclotomicField(8);C;
4 r3 k3 @- w5 u9 \FF:=CyclotomicPolynomial(8);FF;
: K1 B3 @6 f& y+ V0 h
F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
9 `0 s5 [* m3 iQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
) A* ^+ o0 X5 k9 [1 R$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
8 A' V% E, j0 m) R1 e    <$.1^4 + 1, 1>
]
: G6 o- Y1 }# O2 s. X. ^. N6 f5 T
R.<x> = QQ[]
! i& V% \+ k2 @5 `, N; cF6 = factor(x^6 - 1)
F6

6 B8 T& W. [/ _6 N5 O(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
. u" d& B- H( m" u9 S3 }6 p
: t# }( u( p5 Z  t5 GQ<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
0 P. g1 I1 U) [, K7 S& n# J0 fFF:=CyclotomicPolynomial(6);FF;

% |0 o9 y+ L# ^F := QuadraticField(6);
# A2 ?+ H( x% ?9 Q; U+ X/ UF;
  C/ I, j% l6 e, k4 C  Q% yD:=Factorization(FF) ;D;
) C* ~: j5 N( E0 [6 XQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
# C8 T3 A/ }+ N- ~Cyclotomic Field of order 6 and degree 2
$ v& H0 M# }. q* O$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
' j* G2 b: ?' M8 F% a[
& z, ], s0 ~) I  M4 o; }    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
% }; L: }! g+ U6 _& f; u8 [  JF5 = factor(x^10 - 1)
F5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
$ m, V& T3 }+ H5 J( \$ K* ~1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

Q<x> := QuadraticField(10);Q;
: b( M/ X* ], N4 A3 E" tC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

. S5 Q# Q9 p3 I# o" bF := QuadraticField(10);
F;
1 v8 @& ]: n& e( Z5 OD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
% l7 j9 B/ D5 bQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
4 z. P- b, F! N; T( \  h[
4 P/ V8 K( O. i6 c9 y    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123