虚二次域例两（-5/50）

lilianjie

Q5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
; S7 U% K) T, }( p) |M:=MaximalOrder(Q5) ;
/ Q7 L2 l2 u; p# L1 WM;
NumberField(M);
( ?( a/ s; w. S9 I8 D' v1 B* ?# ]" ]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;
IsQuadratic(Q5);
) o; L+ A. b' G) G! HIsQuadratic(S1);
2 Z5 V" F# ~2 S# B" ZIsQuadratic(S4);
IsQuadratic(S25);
" `* @4 \8 w" V5 O7 p, lIsQuadratic(S625888888);
Factorization(w^2+5);  
, K& p. F* L+ Z7 gDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
* O3 C$ u  g) R) R5 d( P/ cConductor(Q5) ;
' a: K, s( F- A7 O4 p, U( ^0 ^
Name(M, -5);
Conductor(M);
  |+ [6 S0 J, `& }; Z) N* m% ^ClassGroup(Q5) ;
$ U: ^' I+ x) A9 i2 c9 gClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
( t( |; v, I. M  A9 c  nPicardNumber(M) ;
  ]0 a& ]& |/ J5 s0 }
* ~$ H$ M, I' H0 B; vQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
) N3 i, @2 }1 m# R2 R0 g" \Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
/ ?0 {$ D$ X1 o5 xQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
; u8 k( p1 }3 I8 pOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
: u# I- g7 E, s- m# a2 m3 ]true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - Q5.1, 1>,
' T7 [& X. ?1 I    <w + Q5.1, 1>
]
-20

>> FundamentalUnit(Q5) ;
, A, f: O3 x- F: H! X                  ^
+ J% G/ y$ O: i$ i0 f' p# X( ^6 \5 [; M! |Runtime error in 'FundamentalUnit': Field must have positive discriminant

) M9 g: _1 x4 `( ?
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

20
8 U  D' h1 X) f2 ?0 [+ _
7 r: ^, i4 F% @+ P>> Name(M, -5);
       ^
, ^2 b# d3 }6 u7 E+ GRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
) u2 X; q. A1 {+ O5 W" I8 a0 Y
1
Abelian Group isomorphic to Z/2
Defined on 1 generator
6 _2 C+ Q1 E) y% p8 o  @Relations:
" q* }' g$ H: Z7 }1 `# k    2*$.1 = 0
3 {* u. i% Q6 ]2 pMapping from: Abelian Group isomorphic to Z/2
& p# p7 u! H; ]Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
6 f. M: T2 c5 h$ S9 f$ F* s1 T# oAbelian Group isomorphic to Z/2
! N0 g6 h( O% ]( fDefined on 1 generator
Relations:
; P) j+ y2 Z3 o7 v6 c    2*$.1 = 0
% I, K) R) Z) b+ zMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
, P+ x$ Z; K1 `# T7 T2 w& z& B5 v    2*$.1 = 0 to Set of ideals of M
+ p( k5 E8 p. |  U" H2 ^3 w2
+ B6 E' ~2 C) L6 O/ W, Y' E: A2
+ |  t: X# w: \( Z: zAbelian Group isomorphic to Z/2
Defined on 1 generator
. H- v, k1 c3 x% e& a# ~" s- wRelations:
    2*$.1 = 0
+ ]9 @  k' i& o/ OMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
# {3 w3 I# m* e; j3 LRelations:
( N' `" T- D0 ^5 U$ l    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
$ U  B; U2 z2 z& }9 J. T# T    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
Defined on 1 generator
% u$ a% z) f: l, Y+ C$ `! hRelations:
5 @4 h* @2 b- u. @5 v2 i7 o    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
2 B3 F2 L% B7 z1 S5 m- q1 J4 [* ]Defined on 1 generator
) Q5 R, w, ]- r/ H/ V  NRelations:
* Z# r* S- p( [# W    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
false
8 m; P& [( C8 \) ?( Ifalse
==============


Q5:=QuadraticField(-50) ;
8 L  |% i# Y3 j4 A+ MQ5;
. L6 ~5 n5 M5 s3 |
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
: h$ L$ ?& _0 b- u! S9 W/ wM;
NumberField(M);
! p5 q$ K& ^; L6 YS1:=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;
  @: s& Z" f1 p) o# e: @IsQuadratic(Q5);
6 G% B5 _3 Z0 u1 j/ bIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
4 i9 Z* I. p# b' jDiscriminant(Q5) ;
, {2 Q: U1 j+ `FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
3 Q& A3 q- g+ S& X
5 w9 {& w/ X2 w5 J6 {4 `( C; ]* oName(M, -50);
Conductor(M);
+ P+ C$ y8 P, J& m" \ClassGroup(Q5) ;
. x  c) ^8 ^; V9 Q6 z8 \ClassGroup(M);
0 b! d' b# S" ~- W9 X0 A1 cClassNumber(Q5) ;
8 A& k* S  Q% }ClassNumber(M) ;
PicardGroup(M) ;
: T! L2 M/ P2 C5 IPicardNumber(M) ;
& @. r9 Q* k% @8 |
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  Z" M  K! }) u% [6 x: KNormEquation(Q5, -50) ;
NormEquation(M, -50) ;
, b- J: ]9 b0 {! J0 a
. t3 ], y5 N) E) [; |' Q& 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
8 j5 ?8 v' _, c+ E6 CMaximal Equation Order of Q5
0 n  o  R7 L2 P: CQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
" ~  ~6 e- u, ^8 v# U* p% Htrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
4 [- \' u- ~: r* Z[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
$ o( C. g8 I: R7 q5 D; L) z-8
" P6 I: {8 d/ M/ a% p5 r6 c
4 I" R% t" ~/ B0 i>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


, C2 _( ?+ `3 E. l2 w% W>> FundamentalUnit(M);
7 V) N% {6 F8 j, ]                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
7 U" s. d3 U% T" s5 ^( j
8
+ L% G+ A& g" s# N9 C% ?
2 }" s: T4 e. [9 V* p, N>> Name(M, -50);
       ^
$ ?0 u9 o( {# c9 DRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
7 Z" U1 N, m7 Y
" Z. k" T% t/ v+ l0 J1
9 N% y6 X( x8 p- ^Abelian Group of order 1
; e: X4 p! |+ @  iMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
  h5 u& y4 t5 NMapping from: Abelian Group of order 1 to Set of ideals of M
1
4 C9 M- n# ?# \2 k, n7 c8 J1
Abelian 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
3 T' s1 {: N3 Z' UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
4 q( m* d. U7 ^" ]Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 S' d' R9 o' P( D. Q% A; G/ v6 B8 v-8 given by a rule [no inverse]
false
false

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;

( s& L' K/ @5 [9 ~& ?- ?% xQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
; \7 W# x& y9 u6 _8 KM;
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;
' g8 H! W, ~  d5 w" y6 e7 NIsQuadratic(Q5);
% a5 Z0 m  z! C% ]6 j( XIsQuadratic(S1);
IsQuadratic(S4);
; k! c8 c) r& t  l7 r: QIsQuadratic(S25);
IsQuadratic(S625888888);
4 C& [' K  d; ?- r& U9 S- i* oFactorization(w^2+1);  
Discriminant(Q5) ;
9 [0 z  p# F6 ^, o- gFundamentalUnit(Q5) ;
FundamentalUnit(M);
* W# o# E- ]( E6 h$ v4 AConductor(Q5) ;
+ @9 |7 j5 F/ J8 ~" l1 F
0 s! t( C8 w" o3 i# f2 d8 fName(M, -1);
Conductor(M);
ClassGroup(Q5) ;
1 S, F. y3 W9 K5 H0 H* LClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
/ s3 ^+ M, U: k# K2 ZPicardGroup(M) ;
PicardNumber(M) ;

2 f# a6 a  s8 t7 E  Z: UQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

) i0 ?# H9 l1 sQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
  m' x0 a1 x* CMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
! n: j' B) Y# mOrder of conductor 625888888 in Q5
8 y% p+ a% }% ?$ D/ `true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
* o/ ]( j" }' a  S( a9 |[
3 s) e% C* D" d0 E# [4 p    <w - Q5.1, 1>,
" f: `9 i: ?  l3 ]# R; C. n    <w + Q5.1, 1>
]
-4

>> FundamentalUnit(Q5) ;
/ [* F' K2 z! w1 u  Z' G0 y                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
" X9 P) ?+ N* r& H. f$ m% B# k2 E
3 I" g0 S/ m1 R
1 @! I! t( q1 z  |5 `  f3 b>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
, h  v, f4 ^  e$ ?. ~0 }% @
4

>> Name(M, -1);
. k0 [8 r: l. h/ J7 `       ^
0 y3 C( k5 p% e+ E% k- M% @' jRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
3 ]( a) _8 V6 L- W$ eAbelian Group of order 1
5 E1 A, U1 b% G% g# @8 wMapping from: Abelian Group of order 1 to Set of ideals of M
1
" M! I9 \6 g/ ?6 M1
- C, K9 m) j+ V1 d; j- H. q9 }Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
( u/ Q$ _# ]" r! U4 C! y" PAbelian Group of order 1
0 N4 G- f6 ^/ I% P+ E3 b6 @! z' MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
! K) ^$ |! w, H# w' a2 ^  w-4 given by a rule [no inverse]
9 Y" z: E: O& w- P7 jAbelian Group of order 1
* Z) b/ E  g; C. V; QMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
, R7 n# ]/ R5 N7 dfalse
===============

Q5:=QuadraticField(-3) ;
% W- d0 V" [1 b0 B5 r6 N% k8 LQ5;
6 B4 l" Z  B$ {" b
' o  b1 Y' U8 O- \& HQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
& q: P! t# v, n/ T( r4 r( S. ?: j8 h% JM:=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;
! @  a8 Y' V, j8 TIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
! {# Z# C+ ]4 n! pFactorization(w^2+3);  
/ Y3 x7 V/ y# N' MDiscriminant(Q5) ;
: _# c+ w' x" g& k/ [& p! K: gFundamentalUnit(Q5) ;
5 z6 Z% k4 {1 q  \FundamentalUnit(M);
/ s7 R" S$ L  eConductor(Q5) ;

Name(M, -3);
Conductor(M);
ClassGroup(Q5) ;
6 Y1 C: g2 Y. Z! ]4 c4 XClassGroup(M);
& c0 b2 z2 ?  v' W8 MClassNumber(Q5) ;
ClassNumber(M) ;
, E, k3 r/ T! u9 m- A" xPicardGroup(M) ;
# o- J% L; j, _/ U, U; u7 bPicardNumber(M) ;

, {  f, [9 }, l. M$ F5 M) t3 XQuadraticClassGroupTwoPart(Q5);
) I( A2 Y0 c& `1 {' K" U0 OQuadraticClassGroupTwoPart(M);
+ I; Q0 Y( R% q# y) ^NormEquation(Q5, -3) ;
NormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
$ ]( k3 L8 W. u" T+ BUnivariate Polynomial Ring in w over Q5
/ T2 P- J! D3 j/ p) l/ AEquation Order of conductor 2 in Q5
Maximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
) j: e1 S" g, I8 W" K1 Z! y  O: Qtrue Maximal Order of Q5
true Order of conductor 16 in Q5
& [4 v" ]! ]7 r7 u' k: ztrue Order of conductor 625 in Q5
5 x+ X9 v" y  c' f7 {$ V- |true Order of conductor 391736900121876544 in Q5
6 E) O: d3 \3 k! J! |[
    <w - Q5.1, 1>,
$ g* C* n  X; @4 q    <w + Q5.1, 1>
7 c; S+ G) Z; o+ Q9 B8 p]
7 r5 N: X; k: |3 G/ J2 A/ U+ o! \, M-3

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8 P1 q4 x/ ~( v! x, \# @- n
>> FundamentalUnit(M);
# W: \( m$ p; w5 o                  ^
# |% M) x2 g  [3 g" A, SRuntime error in 'FundamentalUnit': Field must have positive discriminant
8 D' @6 P: o/ z: L
+ M5 j2 {1 n; A3 I$ c3
6 \% W+ g7 `# W1 Y7 G! s
>> Name(M, -3);
& O; ^2 K" F& n% v( Z( F0 ]       ^
& v9 G9 `5 X- d0 tRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
9 w9 K: g+ l: V/ P# @, j$ I
1
Abelian Group of order 1
5 S( K; ]7 i3 x# DMapping from: Abelian Group of order 1 to Set of ideals of M
8 ?" f: V, n& NAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
1
% K- a# b: @- X; JAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
9 _- L, x2 W( T/ p6 j% P1
: x/ \7 C: G9 q7 w  n) i. RAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% l( w5 e" Q6 r1 i-3 given by a rule [no inverse]
Abelian Group of order 1
9 W4 O0 f; m% G  y2 y0 h, M4 |Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 D* i2 B) j9 H! @6 U3 m6 ^; [7 Y1 c-3 given by a rule [no inverse]
% |8 ^1 b/ ?+ }3 K( qfalse
2 f% s% D9 z  U* bfalse

孤寂冷逍遥

lilianjie

) N. b# E/ x3 {$ D" D
2 X+ u$ ~3 c5 I" a  zDirichlet character
9 j* k' E3 A. n3 M  ?Dirichlet class number formula
; d: b' y' _1 H1 c& @  j% i) \
7 @2 m% T- B3 G7 x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

-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

-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
7 n+ b" p) |5 O
( ?4 `% B0 h* t% |-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，


  W4 u  X- V- U; Y: Q4 \# o  X8 h
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
1 t6 N8 i4 n+ O" J
+ C8 \6 O) J: m2 `
/ @/ }& D" F$ n. y" F& r
' P1 K- S3 J* u-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

% u# ^( h) O7 k
F := QuadraticField(NextPrime(5));
8 {1 X) d+ W0 t1 i' ^% t7 k
  G* n  W' C( O- |KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
: Y3 h% D4 `' s; MConductor(KK);
$ r; D  R- X2 T- h8 p  H% LClassGroup(KK) ;
, `8 ~( l7 @0 y+ T+ c, p/ F+ h4 sQuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
A:=K!7;A;
' Q+ P1 S+ V% \: p, O+ |% r. MB:=K!14;B;
Discriminant(KK)
) C3 B9 R$ {' \9 n1 O9 g
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
2 u; ]+ _$ V- ^28
Abelian Group of order 1
. F; F! h! t4 ?3 P6 i% m, C: kMapping from: Abelian Group of order 1 to Set of ideals of K
Abelian 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 Binary quadratic forms of discriminant 28 given by a rule [no
5 t2 `! r; P. v, P. t! cinverse]
false
5 N! g7 g8 J/ a- ?7
( w# U/ y9 |* Y6 L! o, O  a; Z14
28

lilianjie

1 |8 b0 F3 V6 m: {" M( o! B# d




# h5 ^8 Y# A4 C( B, t

" ^/ T$ b& d, h4 K" d9 Y分圆域：
C:=CyclotomicField(5);C;
1 T) M$ Y6 C, l, [/ VCyclotomicPolynomial(5);
0 C# R8 T1 i% [! l/ ^' e. QC:=CyclotomicField(6);C;
4 E. L: }6 w$ _6 E: v! uCyclotomicPolynomial(6);
* C' f. L" N) @% b! H; ?& jCC:=CyclotomicField(7);CC;
: a. d, a% b) g2 p4 y0 m2 i: |5 WCyclotomicPolynomial(7);
3 p% }  q# t/ Y$ oMinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
$ l; f! j8 `# c% oCyclotomicOrder(CC) ;

5 U; O8 M, K1 d; Z+ C( GCyclotomicAutomorphismGroup(CC) ;
. A% v: B1 c1 z6 ^1 j& k
Cyclotomic Field of order 5 and degree 4
! c& @# z3 g( D3 P0 ?/ h$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
8 I: p/ z8 ^8 H2 L4 MCyclotomic Field of order 7 and degree 6
4 C7 E& \6 \  k+ P4 N6 h$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
; D! J2 M+ b! URational Field
: t) O5 e) j. n: c9 h0 ^: [* C  szeta_11
. Q# [, b  A( ^1 nzeta_111
123
7
( O6 c) ]9 |/ g& d* R% R7
8 t8 O" O9 t8 J! C& @/ aPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
' Z3 K2 ~  a5 _( _) ]. X0 l    (1, 2)(3, 5)(4, 6)
    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
2 n% l' w6 b/ QCC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
9 [3 F) Z2 Q- F. R$ ]Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
5 J: w; [& L/ a6 b% e' PCC

lilianjie

+ ~4 v$ w. Q  K) E5 l! i0 h3 ]

lilianjie 发表于 2012-1-9 20:44
( h9 ]! A0 E1 F9 r, f分圆域：
C:=CyclotomicField(5);C;
3 R" d" M: N7 o" R3 g$ p$ mCyclotomicPolynomial(5);

分圆域：
5 r( A1 ]( P& Y& r2 n* G分圆域：123
1 G  L# O% d1 n. C6 |
( Z7 i4 }: D2 X$ h! n  aR.<x> = Q[]
F8 = factor(x^8 - 1)
F8
9 Y! S4 t) L3 P9 {+ [' t6 I
(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;
4 F, k8 O0 {8 J+ K) PC:=CyclotomicField(8);C;
) B0 \/ e7 g4 g+ VFF:=CyclotomicPolynomial(8);FF;

' q  r  H) j, C1 @: wF := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% _, U$ Z  k0 f- NCyclotomic Field of order 8 and degree 4
$.1^4 + 1
" X) Y' J  }% p0 m  c3 t8 Z# LQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
, p% d# _- I$ ?: @/ t: e; c    <$.1^4 + 1, 1>
& y% i; `: N( J8 }6 V, T; e]

+ q7 t( W4 x( B- C% YR.<x> = QQ[]
F6 = factor(x^6 - 1)
F6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
% E/ p  |' M/ A* F/ v* x% k8 f, nFF:=CyclotomicPolynomial(6);FF;
: k/ n, i) h& f# ~( L
, s. @$ ~; a4 c- c- Q8 l( }: hF := QuadraticField(6);
+ H0 q  ?/ j1 ?  eF;
2 T6 x* b0 a2 I$ t7 y1 cD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
4 [. X9 o& i1 x9 eCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
, g2 _6 d7 S5 z% a" sQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

% N6 l0 s( {& T9 J% q4 ?R.<x> = QQ[]
F5 = factor(x^10 - 1)
F5
+ `  q7 L: s5 ?" k: L(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
2 c4 z* U) S4 u( v6 }; {: z3 r1)(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;
# {6 h. d2 q3 F! a+ N: i2 a6 oC:=CyclotomicField(10);C;
0 _. R2 C8 E! V3 i0 q3 yFF:=CyclotomicPolynomial(10);FF;
- w7 j7 P1 T4 L2 }# K# x
: m5 y% A& C" YF := QuadraticField(10);
F;
D:=Factorization(FF) ;D;
6 J: W- {( \9 M+ hQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
# ~, S& p8 I% j" tCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
. o2 G. h$ W8 d% d9 S( b$ EQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
9 m$ f: v$ G! y! M5 |+ W[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
2 w5 w8 C0 q/ I& s2 a$ n) b]

lilianjie1

分圆域：123