虚二次域例两（-5/50）

lilianjie

5 G$ F6 W# M, e* F8 N
Q5:=QuadraticField(-5) ;
. `- Q& L, x% }2 aQ5;
! f% o! M! f+ D; ^- J8 d' ]9 p
0 k4 {) f* W% N* g' BQ<w> :=PolynomialRing(Q5);Q;
: s. S& Q/ z! r' t" A0 ]EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
( I  g; }" H- v6 _( SM;
NumberField(M);
6 A. H+ ]; m4 D! F4 O, u6 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;
+ |+ {* t$ B# MIsQuadratic(Q5);
IsQuadratic(S1);
3 {/ [. q- V! D. z, i. vIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
: c- b4 b2 }2 [5 T" Y. ^- Q, RFactorization(w^2+5);  
3 |) `, ~1 m  DDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
: g  ^; d: Z  K) Z& aFundamentalUnit(M);
* R' ~6 P5 g, S6 u  S9 d# j+ aConductor(Q5) ;
8 p  z8 v7 \2 v; ~0 h8 q, Y
Name(M, -5);
1 g6 Y8 r) `3 i. g" V, I9 EConductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
. k. i) _$ }+ N# FClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
( s. Y6 c, ], W) gPicardNumber(M) ;

! `: Y+ D/ \' K- R# i; c! N- g" iQuadraticClassGroupTwoPart(Q5);
. z) ?' D, D" I" n" p: jQuadraticClassGroupTwoPart(M);
' u0 w4 z. Z, W( @# FNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
1 q  b' h6 O: E$ h7 {Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
; m, e$ o) H; R: }: q3 S( v& IEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
$ T/ B0 Z2 J' X2 M2 Ktrue 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
true Order of conductor 1 in Q5
, `7 w0 d0 k/ n5 p9 vtrue Order of conductor 1 in Q5
) q' W) v( p) G% Z[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
" M; G5 E. P- T-20
$ J' M  l  L5 A
0 H6 o' n; P7 @' [8 ?5 d' [>> FundamentalUnit(Q5) ;
( ^: I$ V* R: r. m/ P/ o) H                  ^
: V+ j  z0 z. d3 [1 QRuntime error in 'FundamentalUnit': Field must have positive discriminant
3 ?2 K; z' G7 q8 n
  K. \8 A3 {; x  s: K2 O
>> FundamentalUnit(M);
                  ^
! Y: Y1 P7 W$ hRuntime error in 'FundamentalUnit': Field must have positive discriminant
$ ?3 N: J. E- x; y  D! _
20

7 [/ `3 f& m  z5 P>> Name(M, -5);
       ^
, s) Z3 ~- m  [/ b$ T1 U& ERuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
Abelian Group isomorphic to Z/2
Defined on 1 generator
0 x8 H4 l- I+ W) m- QRelations:
) p; S% J. j, F* B; ?9 `" M- {    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
( S5 ^4 [5 g6 W4 O8 v. Q( WRelations:
    2*$.1 = 0 to Set of ideals of M
% [0 H  z1 s: g- |) y  EAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
+ R( s; B9 k. H& |    2*$.1 = 0
* A: F2 C% b& O& @3 U) b0 ], q- KMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
2 Y& j8 \3 M# e$ @: Z# c! hRelations:
    2*$.1 = 0 to Set of ideals of M
2
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
6 ]6 P+ y* G9 K3 }Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
* k6 T6 W+ o+ t    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
9 c$ k8 ?+ k; U; b1 nRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
& E  e. Z% W' C4 a/ f. V: GRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
2 g, X, C* y  C6 Z7 OAbelian Group isomorphic to Z/2
6 K- E1 h7 y+ L$ ODefined on 1 generator
7 D4 ~: M, t* B5 I/ R& oRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
, \  s1 R) Q2 D: J    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
6 h* n1 ~; v( Winverse]
" h# S) p# a4 A; B8 ?% cfalse
6 m* O* p' @' B) u" f" N# ifalse
==============
3 `, D5 v& _. f7 q  V
; y) ?7 Q- ~! _2 B
Q5:=QuadraticField(-50) ;
( _) g# v: v5 I7 _2 ?Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
% G" x: @; F! _1 f8 \; xM:=MaximalOrder(Q5) ;
M;
NumberField(M);
. w0 L/ `7 d9 L3 e4 Y; \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);
/ K! {! w+ D0 f) e* B2 Q1 l" q( YIsQuadratic(S1);
" [& I5 x; a& j$ }- dIsQuadratic(S4);
, U3 \2 I* D( |: t: A/ @1 ^IsQuadratic(S25);
IsQuadratic(S625888888);
3 O- L* F7 ^' R5 I$ KFactorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
9 o- L8 X( O8 k( ~' z- n3 k2 r' ~$ e3 LConductor(Q5) ;

3 T' s* o  ~) O5 |( j- MName(M, -50);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

# E  z) n0 n0 h5 MQuadraticClassGroupTwoPart(Q5);
; h" v- S& P' h% m, G, y) zQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
NormEquation(M, -50) ;
/ d5 p. `$ C& S
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
) o; A0 o, _: H/ BEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
) u4 r7 v5 A# F+ t* Itrue Maximal Equation Order of Q5
2 g5 w5 R/ Z% y6 Z7 Y/ `0 }true Order of conductor 1 in Q5
2 B, f. h. t' i3 d/ B$ O. A# Qtrue Order of conductor 1 in Q5
9 _- v7 S, F+ ptrue Order of conductor 1 in Q5
! k/ [7 Q% V8 G0 {- Z[
! A! H  m9 `/ S: N    <w - 5*Q5.1, 1>,
- |% J; a4 w) @3 D  B    <w + 5*Q5.1, 1>
]
3 s, S7 _+ }# H6 R) ^* D-8
1 t1 A0 r: A- ~
>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
, r; H( P% p0 u$ j8 O  I+ ^8 k1 r! [
1 c3 b! @) T: h5 F1 Q+ S$ @
( w. ^$ a( ]1 ]% R>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8
6 Q% w( h1 Y  V! J
+ B& u8 m* y* U>> Name(M, -50);
       ^
: Y* Z4 Z! S& H2 z; K4 pRuntime error in 'Name': Argument 2 (-50) 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
6 V3 s6 O" ]5 i+ M1 eAbelian Group of order 1
5 B& O6 s* D2 `) u) T' X1 T4 qMapping from: Abelian Group of order 1 to Set of ideals of M
3 S- _! E) K  j! S% D  R. n/ b1
. J6 M8 v; T) R) [& f1
/ \# T5 X5 X3 \3 Q5 KAbelian Group of order 1
9 ?4 S7 d. C6 }" g2 ^, T; EMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
! T; g' `1 J" H; {2 s( u8 nAbelian Group of order 1
+ o5 h, y6 z8 j$ V1 ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
. }# `4 A, o9 r4 ~8 o. sAbelian Group of order 1
* r6 t9 h( j) N. [  HMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
) @+ J& C/ C6 ~" w* W. b: w/ U-8 given by a rule [no inverse]
: q6 m9 ^: t' G! q) }false
false
, E! q/ s7 U( ^; }

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
Q5;
. s3 {) D% A( {4 o
Q<w> :=PolynomialRing(Q5);Q;
( y- V  ^* n) yEquationOrder(Q5);
" A+ L8 F5 k1 |  e( X8 m! rM:=MaximalOrder(Q5) ;
M;
  G; g0 Q& m5 w9 D9 r! E3 [+ XNumberField(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;
) u5 Z! ?# A; Y: Q7 H1 R; S% `$ EIsQuadratic(Q5);
) |9 v" z# {, i+ W& LIsQuadratic(S1);
IsQuadratic(S4);
$ N/ T1 q/ N0 }" G. r* I" R4 vIsQuadratic(S25);
( f0 L2 G* \6 T  V, ~2 z% ZIsQuadratic(S625888888);
) y" q8 t( Y+ Q$ ZFactorization(w^2+1);  
" s# k' q: m, v) HDiscriminant(Q5) ;
- i. l3 W. Z& U- J  iFundamentalUnit(Q5) ;
* y" l4 b; ]' p) aFundamentalUnit(M);
Conductor(Q5) ;

0 I% K: V: F$ E: |Name(M, -1);
Conductor(M);
1 X6 D* m( t) u8 a" K# ?2 w" OClassGroup(Q5) ;
- D* b( p" L) I2 `" V! mClassGroup(M);
! L# q& E8 d& b/ e( XClassNumber(Q5) ;
. [/ e7 O; k) C$ a# qClassNumber(M) ;
PicardGroup(M) ;
* \: [& A$ v4 Q  N$ I6 K. V+ VPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
2 G/ s8 a2 h- t6 HQuadraticClassGroupTwoPart(M);
1 T$ s3 e: l# u! G* f! u1 [NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

- ]* c) f: C6 ~8 `* R' }Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
; r- Z4 \3 R" S5 g+ k2 |  f6 I& [Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
6 |% [0 c) M0 F. KMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
8 E7 s  n6 M( Y8 h7 }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
2 C2 c% I* `# m% S+ ?true Order of conductor 1 in Q5
' g6 q8 D0 i/ p* Xtrue Order of conductor 1 in Q5
' e* T4 B9 O4 _/ Y[
    <w - Q5.1, 1>,
7 |% ]% e! ^/ s# j8 D) `    <w + Q5.1, 1>
]
-4

) p6 |; R. o8 c: |* _3 s>> FundamentalUnit(Q5) ;
                  ^
8 }0 I% `8 p% {Runtime error in 'FundamentalUnit': Field must have positive discriminant
6 a+ r! q1 |4 a$ [) |+ Z, i

>> FundamentalUnit(M);
# t% f  Y0 |) c2 X" f. c6 R, ]                  ^
. i$ P6 l0 m4 s8 ZRuntime error in 'FundamentalUnit': Field must have positive discriminant

3 _+ y0 }4 r; r: \, I* q1 e4

>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

5 D8 [$ \% N6 e; f, f* U. T) q1
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
& ]% \# n3 T& J: ?. x/ P1
, q" c3 O* ^. M; v& B4 b1
) f4 i& k+ c: X1 {( e8 O5 q$ yAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
3 E" `! N, P! _1 D1 x9 Y0 n1
Abelian Group of order 1
6 [! k* I4 Y' ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
3 b: s' `4 F) Q1 D-4 given by a rule [no inverse]
Abelian Group of order 1
2 c. d; U4 ?: N) H8 ~4 gMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
( ]  V' A3 p) E! O; x: w* h* ifalse
false
===============

Q5:=QuadraticField(-3) ;
4 g; q; S( J7 jQ5;

5 m5 k% `9 e& E5 \2 T# C5 Z! ?Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
3 U" d2 w; y" v  UNumberField(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;
IsQuadratic(Q5);
. F# r3 u4 W% g9 Z. w) V; I9 ]IsQuadratic(S1);
" P% d6 J  h6 kIsQuadratic(S4);
. F9 P% d6 m/ Q  a# ]) n3 N5 r: i  E/ |' rIsQuadratic(S25);
& i/ b$ B; f1 z+ D/ i, pIsQuadratic(S625888888);
Factorization(w^2+3);  
Discriminant(Q5) ;
1 L- f# g: J! L5 d4 x8 B$ L; |FundamentalUnit(Q5) ;
: O/ K; o  a) OFundamentalUnit(M);
Conductor(Q5) ;
5 K% c: E) m; `# R* B0 H0 i* T9 X
1 v( V( Y/ c" ^9 H# k. V: C& v3 rName(M, -3);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
( l  A- O5 r$ G; qClassNumber(M) ;
PicardGroup(M) ;
& `9 M+ c! _* Q7 T9 a( UPicardNumber(M) ;
9 n9 C0 Q2 |. k4 z- o
( R6 t; R% C+ X7 P3 zQuadraticClassGroupTwoPart(Q5);
% Z1 S- l3 R; AQuadraticClassGroupTwoPart(M);
6 G9 D4 l/ N5 PNormEquation(Q5, -3) ;
. C1 O+ ?  `" \/ N* E$ fNormEquation(M, -3) ;
: J7 |+ f! N: r# V" m2 X
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
- b; U8 x* r5 }3 g' JEquation 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
  l9 Z: [2 q! j8 ^true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
3 ^/ C% j) |! E  ~true Maximal Order of Q5
4 x8 o4 c' X% m) j7 F7 F/ X2 b  c' atrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
( p* n, J  R* x( j1 b[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
7 B7 R5 c, K" K( c1 e-3

  B" k5 A, q4 i8 M% i- Z4 ^- s>> FundamentalUnit(Q5) ;
% w$ u0 L" O/ h* \. X                  ^
% s  x; X0 O' l1 xRuntime error in 'FundamentalUnit': Field must have positive discriminant

6 J6 n$ i9 d5 v/ g4 S* i
' N. o8 H# z; b" O>> FundamentalUnit(M);
                  ^
' b1 E% R2 I1 M2 R/ aRuntime error in 'FundamentalUnit': Field must have positive discriminant

% c. D; S) ^! f! y* E" F* Y3

>> Name(M, -3);
/ _0 V3 U' P5 B7 o5 S0 n       ^
4 ~2 q1 H0 r# i  r2 p$ t: G$ @Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

, g& V4 M, O+ e" `1
! _2 i" ^0 F/ X1 W5 Q4 ?* r; aAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
+ b+ \& }2 Q" k9 R( ?1 S7 z1 e6 XAbelian Group of order 1
. k  y: [0 {6 }/ W- }1 m1 y  aMapping from: Abelian Group of order 1 to Set of ideals of M
, S4 [% c! G# u5 i1 D$ f( p# o6 t9 [1
1
Abelian Group of order 1
6 `/ r* C7 X; a: N  W4 f, ~  }/ wMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
$ C3 J2 }1 z7 e3 B: |9 G1 tinverse]
# D* a' o& H: d2 p! t1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
- r/ W9 Z$ B# c$ @+ {" F$ V+ Y-3 given by a rule [no inverse]
" _! b! k; Y+ X8 R7 O  B/ @false
9 s! [/ E5 y; Y) K# S$ pfalse

孤寂冷逍遥

lilianjie

Dirichlet character
* `  }7 e7 g/ wDirichlet class number formula

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
  B7 Q$ F( C6 t2 l! x
-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
1 R9 s" @' C& H. ?$ y8 P. uh=-6/（2*3）*Σ[1*1+（2*（-1）]=1

5 M) S: n# P) U  S; R-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，
: X+ `0 L  d* q1 j9 f3 w4 ?5 Q; u


h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2

- b% |$ B) w5 d. d* `
, Q5 y# A* U# d& a) l9 C
-50时  个单位根                          N=200
% M4 b9 @( |+ l( Y6 v  e

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

, M5 v3 `- R' |: k) b2 W( q. i3 P) i
F := QuadraticField(NextPrime(5));
$ p7 D! L1 z, D3 G/ p
: ?$ h; C: o; L5 B, M* A- fKK := QuadraticField(7);KK;
, J) A' f5 f$ P7 \7 l( yK:=MaximalOrder(KK);
Conductor(KK);
ClassGroup(KK) ;
0 s' _2 [; m& PQuadraticClassGroupTwoPart(KK) ;
2 U/ d9 `$ s7 A8 z: y1 {NormEquation(F, 7);
4 F2 S" {0 [3 p5 T5 sA:=K!7;A;
3 {0 _' ]5 G5 v/ XB:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
) A$ J: a# v  o; Q4 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
+ x8 S2 A. Q0 F, XMapping from: Abelian Group isomorphic to Z/2
$ p  H& t* h$ d5 O3 [! RDefined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
14
! n: ]( x# E- o" {/ s, L, O28

lilianjie

( l, l8 H' \2 s* [! c




  s  S7 L. h0 \! B, n
( n7 x" C: h9 H. s- x* e$ i分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
! p$ a. D+ A9 R: e# M* TC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
* @9 \' D6 w: e* sCC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
% F7 m* A# E  Z9 X0 WMinimalField(CC!8) ;
MinimalField(CC!9) ;
: T0 ^! l- Y! }, [6 F% LMinimalCyclotomicField(CC!7) ;
% h6 g( i+ Y" h: J5 w0 {RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
Conductor(CC) ;
% r, j2 w8 u/ r. @( }  MCyclotomicOrder(CC) ;
$ \& E' Q4 p+ }* f' H* G$ R
CyclotomicAutomorphismGroup(CC) ;
% I8 I8 s+ g1 ~; R; q: Q1 r  }
Cyclotomic Field of order 5 and degree 4
4 h! [6 u4 q# X- O  U" b$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
  {/ x, j# S, H- @/ Y. _4 }9 Y$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
# e3 Y4 u3 f  G4 ]2 _) w2 b- ^Rational Field
Rational Field
- P, z, h) @& ]. m8 {* NRational Field
7 W0 [7 A  l8 xzeta_11
zeta_111
123
) r1 d8 D+ X1 q4 A7
7
" I, m% x. h3 ?7 d/ T+ }% j2 A5 A5 T7 PPermutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
, X# K) `7 C  S4 @9 o    (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
CC
# D5 Q, A& m2 T3 a# ^3 C. QComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

8 }, p7 Z* l2 j- l. C

lilianjie 发表于 2012-1-9 20:44
分圆域：
C:=CyclotomicField(5);C;
/ v& @' O0 {8 p' L& T( f$ ?1 bCyclotomicPolynomial(5);

/ v+ i2 R6 ^1 C9 ]8 I' F
- c9 u( j" w' d+ l0 N/ \分圆域：
3 Q5 c# y5 n. Q! C分圆域：123
3 E6 Q# Q/ k" g# [4 |
" H7 U3 Q! K" j; UR.<x> = Q[]
6 I1 `7 C4 V. d3 E4 BF8 = factor(x^8 - 1)
1 w8 f. A. h/ G+ S# e0 \8 [5 G1 {3 _  jF8
$ `9 r4 q6 x; a5 t
1 |5 z! b8 E4 S5 C+ U(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
) A3 y& h. \4 k" s% P+ n1 v5 L
/ `3 G: {- m8 z) G6 [Q<x> := QuadraticField(8);Q;
% u4 A# e2 A; I' R! q6 H( JC:=CyclotomicField(8);C;
7 Q! ^- U! C3 ]FF:=CyclotomicPolynomial(8);FF;
1 I* l- z% s  K4 @! D0 Q% P& G
$ a& n+ r- N7 q2 mF := QuadraticField(8);
F;
& k. W. j/ t/ {D:=Factorization(FF) ;D;
: B4 W) S9 F$ L& k4 y& Y  DQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
( S3 e, h% b1 S% B2 G1 y$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
% }4 M  Q; o6 \5 Y  `* P; a4 ~! @[
/ \8 k6 V- {. C2 I8 H    <$.1^4 + 1, 1>
]

7 L  f; B( M1 X8 VR.<x> = QQ[]
( N* E- v9 [7 sF6 = factor(x^6 - 1)
F6
# r: @- ~0 ^  N7 V+ u
' F; D5 D( i. K7 [, V( X/ e(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
  n' u6 a4 E$ V: |+ R7 s
Q<x> := QuadraticField(6);Q;
/ {% D+ \5 K+ i! S9 HC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;

, Z$ e1 {. Z) ?' }1 }' |; BF := QuadraticField(6);
F;
5 W' j1 N3 a6 G$ @D:=Factorization(FF) ;D;
1 e* Z9 J& z# cQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
, a' h6 G3 q( D% R( L[
    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
3 U* U) k; q# V( q. W! e8 |F5 = factor(x^10 - 1)
F5
% G# I+ e! Y! W5 @9 r(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
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;
C:=CyclotomicField(10);C;
8 D$ v. d) {& e3 l. e7 JFF:=CyclotomicPolynomial(10);FF;

8 a' g, V0 a  P) O- KF := QuadraticField(10);
1 n2 v; a' v" Z0 u0 _F;
( B, S; Q6 t2 X1 W1 P- m+ q$ XD:=Factorization(FF) ;D;
" H' a. x/ t+ y' gQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
5 x: d: I, I1 [Cyclotomic Field of order 10 and degree 4
. L9 J2 r, C0 y$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
0 o+ Y: @5 |3 V$ j9 H% u# ^    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
+ N8 P" a) N: ~: r3 ?& U6 F]

lilianjie1

分圆域：123