虚二次域例两（-5/50）

lilianjie

8 O9 `2 w8 Z; K, c' M$ X
% }9 z& u6 [6 r0 O5 A3 M& E/ J7 MQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
, A% D& i; F( y# }EquationOrder(Q5);
8 B( j2 o2 d9 t* G2 VM:=MaximalOrder(Q5) ;
M;
NumberField(M);
5 [+ p6 |5 s3 X+ ^; h2 Q  G6 O+ mS1:=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);
IsQuadratic(S1);
IsQuadratic(S4);
, s2 J, P( h: ~IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
) Z7 W! B7 j4 o' W/ J! j# \Discriminant(Q5) ;
; D4 o  Y& c; L$ g& D& ]FundamentalUnit(Q5) ;
/ z. T* Y. J! @6 i; n/ ~FundamentalUnit(M);
' d: W; y2 \7 L! j) j: FConductor(Q5) ;

* _6 W8 i+ g* U5 P7 X, a$ TName(M, -5);
% w2 \; n# m6 K& H) D* RConductor(M);
  N; M+ M% i$ aClassGroup(Q5) ;
ClassGroup(M);
; J) `9 S- t# c  r; mClassNumber(Q5) ;
8 @+ i# J# p$ J. L/ O* [ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
2 J) E4 Z8 c% Q' A2 F0 o# z9 \0 kQuadraticClassGroupTwoPart(M);
; K& \* @$ j: j4 g# d" }0 pNormEquation(Q5, -5) ;
2 M, O& h0 T" N& g) g% ^. {NormEquation(M, -5) ;
$ Z+ d; m$ {3 @+ DQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
# c5 r: s$ X9 Q( Q: g  vUnivariate Polynomial Ring in w over Q5
6 e7 \5 h! P3 V- LEquation Order of conductor 1 in Q5
7 x) r( p0 |6 S& fMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
) t/ o" n; h6 c! c' U& {! V( D7 itrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
+ ?$ ]1 Q3 e) e' e, N% @2 T. q    <w - Q5.1, 1>,
% a1 \7 T/ \, \% e! b" a1 X    <w + Q5.1, 1>
0 q6 v4 _! Z" U. L1 s& L& q9 g- ^]
$ w" Q6 ^' t0 L5 o# {-20

>> FundamentalUnit(Q5) ;
                  ^
& j( x+ {4 r- O. [( xRuntime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
; y. a9 i7 S' @5 y) J( y
5 \- {' C( j; I2 c) U' C$ r$ X( x0 [20
/ F! S8 J0 h: C& r# Z+ _
>> Name(M, -5);
; ]# I$ t' E/ V6 J7 N       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
, `% M/ Y. T9 BAbelian Group isomorphic to Z/2
8 S' v, \+ O! i- ?Defined on 1 generator
Relations:
+ V; H) V2 l# Y6 e* L& C    2*$.1 = 0
/ z) p9 {* r) s2 l: q; D4 zMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
4 p6 n# \1 B, |9 N# M; URelations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
0 ^. _# X) n! }8 q* l' w% u: XDefined on 1 generator
4 v, |+ R5 ^5 ^& a  u( R5 i3 a! hRelations:
' j( C+ @9 B% Y3 [) L# W    2*$.1 = 0
0 l; H. F+ e) m) c  O' X9 UMapping from: Abelian Group isomorphic to Z/2
1 F' J/ H: m1 S' F* tDefined on 1 generator
Relations:
6 v- P6 T6 R: H( I' ~    2*$.1 = 0 to Set of ideals of M
2
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
8 G1 ]$ a$ F& x. v! d% h# VRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
6 T+ H2 E! ]0 r5 uDefined on 1 generator
, N, L% Y5 {  N! W( q% i2 aRelations:
; ^- _4 ?: }( s4 s4 i& [    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
  {6 x4 |& P1 K% _4 o' b: l- W; }2
. B0 L, H2 t- |Abelian Group isomorphic to Z/2
2 t' H7 ^# u% e' f1 s. Q( DDefined on 1 generator
Relations:
0 O- _2 Y$ I0 d* S( H4 _) j' V    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
- i( n; r, L: F) v8 TDefined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
' C% J8 g, y8 P7 x* w! yDefined on 1 generator
Relations:
7 ^  t0 M" \' c; @/ W- P    2*$.1 = 0
3 m! o' h* ]# o0 Y- RMapping from: Abelian Group isomorphic to Z/2
, \0 X  p2 M8 M# Z5 ~Defined on 1 generator
8 U& T, J1 ^; t% H. f3 ^. O4 h2 ~0 {Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
# U! y: q/ t" ]; a1 {/ rfalse
false
==============
& u; p3 K" D' w+ V! c+ o5 Z
( N3 I8 q: F% ?( k( l6 K6 H$ p$ s
& [3 ^" n  Y- C8 E7 i! mQ5:=QuadraticField(-50) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
, X" b. R! D, ^; XEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
$ `7 P( D8 _7 e: g* i! nM;
% Q+ x! @" O2 k% N" ?' w( ENumberField(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);
2 O8 [; \- ~( m4 F0 @IsQuadratic(S1);
IsQuadratic(S4);
' N) x! h1 y! T+ X$ O- b) ^. W: A& qIsQuadratic(S25);
0 g/ S  d- s) D6 K+ T+ VIsQuadratic(S625888888);
7 m7 l  V5 ~! Y: I( u; L9 r- qFactorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
3 B" @' O4 [  G# v" fFundamentalUnit(M);
Conductor(Q5) ;

: |8 O( K4 \# w( j0 e/ ^7 RName(M, -50);
Conductor(M);
ClassGroup(Q5) ;
0 M5 e4 D9 Z, z& L9 w; HClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
+ J; K. N% Z  S! W! O' V
) G9 u- v( D, Z2 ~# d, UQuadraticClassGroupTwoPart(Q5);
! V* m+ ]1 x# FQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
NormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
0 @" h" G5 b# P  sEquation Order of conductor 1 in Q5
5 b  P* j- M& n1 W( Z" ]Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
6 B5 L+ k: B8 w; V6 J$ l- ^8 M' NOrder of conductor 625888888 in Q5
7 C3 `! m3 i* C/ O8 mtrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
5 @2 C8 Z5 _2 ?; ftrue Order of conductor 1 in Q5
2 {6 ?( |" @" atrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
2 t$ z. N- j# V; T' W7 X/ H- R) X, T    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
]
-8
: M7 b* o1 B( d: `& v  V3 G0 T
>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
7 i: D% r2 d4 D5 d0 p$ J+ N' X) F
4 t# T6 _; I. _. m: I2 O
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

8

>> Name(M, -50);
# @- R, I) E: t' R. }, E% ~4 X* S       ^
& w1 W1 Z5 \7 d- }Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

6 N' C  @$ ^( t* I4 z+ g1
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
3 n, G+ I4 A  y! o0 N1
Abelian Group of order 1
1 B0 m% q8 E# Y' u$ h: j) ?Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
: F, h( a7 P0 R# xAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
) n5 j2 d* `4 d8 x2 |Abelian Group of order 1
8 [7 z# k! a# u7 R1 N& G, YMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
+ o8 [! r, h9 Y4 w6 z7 i# C4 rfalse
false
5 n& W% ]) r( e+ r4 @

lilianjie

看看-1.-3的两种：
- w6 M# `* y8 g; X% C  U
8 v, ]1 q# o$ u* x2 h4 rQ5:=QuadraticField(-1) ;
Q5;

' e7 V1 I; Y! {* B0 rQ<w> :=PolynomialRing(Q5);Q;
& p9 k8 X7 E3 _9 Y; o" ~& {' ]! ZEquationOrder(Q5);
* R3 u  G  a4 Q: g  ]M:=MaximalOrder(Q5) ;
- t2 E3 T% D% W; q+ E2 yM;
NumberField(M);
. r2 j" S: W; ~! S' f2 WS1:=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);
1 }+ D7 z" A+ E+ ~/ H5 ^$ JIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
3 g! H% c( I1 d6 ^# N; G" dIsQuadratic(S625888888);
Factorization(w^2+1);  
8 \6 m5 l. `, W* p9 `5 R% ADiscriminant(Q5) ;
7 Q4 a. ?8 D# @- m7 P& l* P' TFundamentalUnit(Q5) ;
% s; i3 ~! S$ _7 I) U% O; }FundamentalUnit(M);
Conductor(Q5) ;
$ Z" v, Y( J% R3 Z3 y( a
0 z3 d& k& L" O- i- ?& a3 H" tName(M, -1);
Conductor(M);
) W( r( s$ c. J& M; k2 y8 eClassGroup(Q5) ;
ClassGroup(M);
- J5 Q5 b+ T0 s) P3 ~, XClassNumber(Q5) ;
0 r6 r" m/ T: [& LClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
9 j! ]3 {% E0 sNormEquation(M, -1) ;

$ N2 h, E& ^( ^5 UQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
. f' i$ Z  B+ |/ T3 RUnivariate Polynomial Ring in w over Q5
: [0 P3 C3 T) x# W- EEquation Order of conductor 1 in Q5
& u: z( J' @# y- N2 N) d' ?Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
( r! G' E) C3 u/ M4 d1 y& K# ltrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
* D; y5 ~3 x1 a# Qtrue Order of conductor 1 in Q5
7 }9 T' [. Z/ ^true Order of conductor 1 in Q5
[
: x3 s8 L8 {3 V) }$ P/ C/ Q( S    <w - Q5.1, 1>,
    <w + Q5.1, 1>
3 R: d+ y4 y/ M8 M7 p- `]
-4
4 S0 V$ ?, V) R3 Z& ]& U! L, \9 R* w/ O
. z8 j$ w* l8 O& [9 n# ~  D>> FundamentalUnit(Q5) ;
+ Z( i7 s9 A5 }" Y3 @                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
" y& z7 ^, V0 _8 x5 U" H7 L

>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
3 |' l. x  P6 x
1 D! M9 p$ D# h/ X, ^  S4
" M# o+ L/ t' q7 o# v
2 i; f, A# ], R" T+ b  S2 G>> Name(M, -1);
  @% V, v" [; q& ]       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

1
" k/ P8 e; R/ S0 oAbelian Group of order 1
& [: V9 I9 m& A& z. q% U& e4 DMapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
* g7 M& R, u2 N) j, @; DMapping from: Abelian Group of order 1 to Set of ideals of M
1
5 d" ?$ p- n- A0 Y; z1
6 R; d  @. t) l! h- j" MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
+ B  w( X4 u, Z# {; Y* C$ EAbelian Group of order 1
  w& g" {" x; MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
! c% _/ N8 g0 Y5 [# oAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
. y; @/ r; m& q2 ~; Kfalse
false
6 t/ Z' k, m% T( L===============

Q5:=QuadraticField(-3) ;
Q5;
# e, {: H- C% v5 ~, n
Q<w> :=PolynomialRing(Q5);Q;
' K7 K: _  \9 F( \6 Y; rEquationOrder(Q5);
" E5 R$ i) }; G& t1 @3 QM:=MaximalOrder(Q5) ;
M;
NumberField(M);
" |( c$ E& [# H8 C2 E4 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;
' m) l- M* A8 p% aIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
* }  T2 s( h6 N' g& h$ nIsQuadratic(S25);
1 L) }( F& W) GIsQuadratic(S625888888);
& r% U4 S% r/ b' vFactorization(w^2+3);  
! [) L3 l* O) K$ N7 U: q, HDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
1 N' H: l' {- r4 q6 J7 c9 Z9 vConductor(Q5) ;

8 }. m8 [2 @" _/ g. y( oName(M, -3);
! g  R" N! m* C% zConductor(M);
ClassGroup(Q5) ;
( ?; g! N7 m7 pClassGroup(M);
* C; L& X: S* S* RClassNumber(Q5) ;
0 b0 H- A' G8 `) p, s- ?ClassNumber(M) ;
PicardGroup(M) ;
$ ?' }6 U3 e: S/ u  l' ^4 d! UPicardNumber(M) ;

7 ?2 E7 v7 \$ u: j+ {# nQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
/ P; z# B; T  h" W- \$ _NormEquation(Q5, -3) ;
1 g1 R$ u1 ]5 JNormEquation(M, -3) ;
$ J4 W1 c% L5 q
8 e: t& c' a9 t5 g5 W! sQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
' m3 o( h& }% b. N$ _; uMaximal 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
true Maximal Order of Q5
true Order of conductor 16 in Q5
% e) _) s; L, D8 [true Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
" S* R2 b, t, [; L0 z( s- I[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-3

>> FundamentalUnit(Q5) ;
                  ^
& a2 g) ~, C, I$ r, S4 K9 IRuntime error in 'FundamentalUnit': Field must have positive discriminant

$ l, f" U! o3 g% Y) l) J
>> FundamentalUnit(M);
5 H. Y2 w2 e4 v2 T0 u                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
' l0 B9 ^* |  I& V5 o; A
3
2 g) `. d& G. B0 B$ H) o
>> Name(M, -3);
  {2 o1 w! z' S- A7 f$ a3 D. a( T# T       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
2 U( g! C$ a: C! n! X7 Q. X
6 `! H  S  A4 L7 c% t1
Abelian Group of order 1
9 z5 D+ u; j) ^0 G1 A4 y2 dMapping from: Abelian Group of order 1 to Set of ideals of M
$ e' O+ _5 H/ a, BAbelian Group of order 1
! D5 M/ P' M' A! d$ R3 a4 xMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
, B  c% T7 Z, ?Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
2 W9 v% O: s" G- y1 x5 ?( ginverse]
8 v4 f! i$ m6 {: m" L6 v- l- J1
) v9 [+ N! C/ L" ?Abelian Group of order 1
7 m2 x1 E! K5 B3 Z, t, bMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
, ]' m, u$ v1 G-3 given by a rule [no inverse]
Abelian Group of order 1
$ r' ]0 G" e  R, g) C  G6 @Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
# U" B* |. \* t6 I. ^3 ~( N-3 given by a rule [no inverse]
' z, @* _8 }( q: X! g* q$ V9 y; mfalse
. S* D/ m' I" j' x2 jfalse

孤寂冷逍遥

lilianjie

6 |0 x& d7 {6 G8 H. D* [
Dirichlet character
. V7 _; Q! m7 QDirichlet 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

3 d8 k0 Y8 z$ T-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

! m) O/ Z: b9 J/ g/ r4 R-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
) T" D4 L" x% w% `9 F' z* M/ `h=-6/（2*3）*Σ[1*1+（2*（-1）]=1

-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，
- \5 d) a; j4 U/ ?, Y  b
9 D% G3 [* ?/ L) H5 n) k
2 r4 N9 N2 ?9 d7 I+ {. H+ N
' y7 s& j; `, Lh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2



-50时  个单位根                          N=200

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

* w8 x  d7 b% j  Q
F := QuadraticField(NextPrime(5));

- v7 t' R1 O, G5 w3 yKK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
6 [0 p, a2 g" YConductor(KK);
' \4 X1 i6 J) hClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
NormEquation(F, 7);
# E9 K' [# `* o, R  OA:=K!7;A;
B:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
8 N9 |: a: ^  {7 h) e/ J; Z' F28
3 u0 }! A, {& g9 i8 {) {Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
! i9 }( P5 x( Q' \7 t2 h1 }& UAbelian Group isomorphic to Z/2
. ^9 `3 P+ }% X/ P0 }4 K- hDefined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
) l8 ?4 Y4 B5 X, vRelations:
+ \; S8 a5 S, m9 k$ R. W& f    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
0 e0 j& S3 ?" L0 Winverse]
false
7
- C4 Q& l( ?3 {14
28

lilianjie

  p3 B3 D* G2 e, F% q

! b7 p0 k  o& K$ C3 l1 Q9 g
0 v# {! ^$ v+ R2 f4 m
, K: Q  Y" K2 z; j

3 X% _+ j) v: [3 U# m
! V# P/ D3 H& }$ S- [: Z( J" X: @' G分圆域：
- y6 f5 Y3 m; u2 M: z$ rC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
) Y: G4 P% N9 S: ~  pC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
MinimalField(CC!9) ;
+ c5 I* `' u4 \, QMinimalCyclotomicField(CC!7) ;
6 Z) @' C: K, u1 S5 ~RootOfUnity(11);RootOfUnity(111);
1 G1 Z8 G0 b) Z7 X5 p$ uMinimise(CC!123);
5 X* ]8 d: r4 {5 A. n0 HConductor(CC) ;
- n$ S% l( u3 y: {) X: b- i, G- TCyclotomicOrder(CC) ;
& }  q9 l, f; s& S0 c: X3 _4 A
CyclotomicAutomorphismGroup(CC) ;
$ d5 E- e( S" T
0 S/ \; T5 ~8 x. u  ]  K+ @7 e, s3 ]Cyclotomic Field of order 5 and degree 4
! Y, T" G# B/ M* ~3 I3 T$.1^4 + $.1^3 + $.1^2 + $.1 + 1
Cyclotomic Field of order 6 and degree 2
3 S3 v8 H4 ?4 \& p/ L$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
6 C  {3 m3 I2 l$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
Rational Field
Rational Field
8 A2 ?- x  h7 ]( C1 w# vzeta_11
zeta_111
+ a( X( |. F5 Z. A. \! _" u123
7
+ G/ a1 [0 z% A2 U1 `. j: h% P7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
" o9 u- J3 M$ D# a% l    (1, 3, 6, 2, 5, 4)
4 v! a) A+ ?* _7 O5 ]7 P) A3 fMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
0 l3 b, Q, F3 \4 V" }Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
9 O* X6 f) R& l  i0 R: oCC

lilianjie

lilianjie 发表于 2012-1-9 20:44
分圆域：
7 x* x) O8 s6 G! V- }C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

) b. c% }) g; ~" ~& [分圆域：
# k9 z0 [- A0 K分圆域：123

R.<x> = Q[]
F8 = factor(x^8 - 1)
1 z- J9 Y1 q0 |" a4 tF8

) I. r7 j! r+ e( f. w% C(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
4 U# w4 L7 v  d8 J# {$ g
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
3 q, E6 U8 S! u$ fD:=Factorization(FF) ;D;
  I* M& D6 K$ lQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
! w: x" X$ x4 a& NCyclotomic Field of order 8 and degree 4
$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
& w8 n% N" ]; c& ^1 l5 x4 \6 j; h- }( N' W    <$.1^4 + 1, 1>
]

/ L+ `% Q' H+ _0 }) ?; ?7 UR.<x> = QQ[]
; ?$ L. ^) |4 l; n4 xF6 = factor(x^6 - 1)
% d+ n6 [1 A! _* O! SF6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
) e+ `5 O5 N; j
6 C$ T# T, i0 p* I, LQ<x> := QuadraticField(6);Q;
4 V  r0 \; ^6 h- mC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
, {' |: ?6 p: h+ m' O
! F/ W+ p$ L2 pF := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
1 p/ h# a6 n! s! a9 aQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
, K6 }0 t4 ^+ \, h1 M$ YCyclotomic Field of order 6 and degree 2
8 v; r3 t/ t7 Y. _0 X) l$.1^2 - $.1 + 1
+ {% m: v1 u0 G% q! r& ?Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
, q. q5 f7 Y% P; W* U) w[
! `, Q( n) m2 a0 ^' F    <$.1^2 - $.1 + 1, 1>
]
6 j4 p+ _9 L) X
R.<x> = QQ[]
' c+ _! u& C: L; u9 _F5 = factor(x^10 - 1)
6 d7 z: @8 c' F1 OF5
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
8 w6 L/ M& z$ {, L6 b1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

6 v3 N: c& b/ v4 R, z& uQ<x> := QuadraticField(10);Q;
. _9 q, x( Z% _4 t# uC:=CyclotomicField(10);C;
, ]5 e/ l8 y, v9 Y0 y  W  O) }5 iFF:=CyclotomicPolynomial(10);FF;
: Y) x7 J. F: j8 B7 \& Y
F := QuadraticField(10);
' ^2 A7 m2 k+ s/ Z/ [' RF;
; Y/ M; F' w' j5 kD:=Factorization(FF) ;D;
- F' ]. o) M  g, f! g1 sQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
2 C/ e. Y2 m  k0 |% x$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
4 ]) K$ t+ s% j2 m' @]

lilianjie1

分圆域：123