虚二次域例两（-5/50）

lilianjie

! r8 R* R+ D  b7 O) O! M
2 \/ E- A& N0 m$ x% _% VQ5:=QuadraticField(-5) ;
; z3 ?" D/ a/ @$ o- W( Q5 h; WQ5;

Q<w> :=PolynomialRing(Q5);Q;
, e6 q) H" Z5 zEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
% }7 S) L7 _1 r3 L) i3 HNumberField(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;
6 Y( n8 ?0 u8 A  A( HIsQuadratic(Q5);
3 p( i8 p3 s, P$ s8 z, TIsQuadratic(S1);
IsQuadratic(S4);
2 ~  l  f3 t& A' s* y- f2 V; MIsQuadratic(S25);
IsQuadratic(S625888888);
( ^1 G" k- Q& s6 x: a$ R$ nFactorization(w^2+5);  
Discriminant(Q5) ;
; g/ j; Z0 b" y9 RFundamentalUnit(Q5) ;
3 |1 Q8 q2 j5 @* j1 WFundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
Conductor(M);
5 j0 Z0 v) I9 n8 Y7 \8 j0 I/ N! ZClassGroup(Q5) ;
ClassGroup(M);
8 @% h) h- D$ j6 m& nClassNumber(Q5) ;
ClassNumber(M) ;
+ N1 q/ K- J' }' t- E" m/ hPicardGroup(M) ;
5 _" a% _1 R3 Y( A. l! KPicardNumber(M) ;
( d0 Z4 @9 V- {; S# L
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
8 x/ u1 L7 n' f4 q9 A; i4 BNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
" e/ \/ }; Q) i9 u3 JQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
/ T) ]" V+ o* u( `2 ^- FOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
# N. {8 k; @* S' n4 c[
    <w - Q5.1, 1>,
% J3 W$ a3 |4 Q& m5 t9 K    <w + Q5.1, 1>
]
# ~8 I9 S: y: a+ a; B-20
; a& P: }1 W# V+ Y
5 J: V& `% T- ~: u- L1 O0 X>> FundamentalUnit(Q5) ;
3 U8 D0 b: P1 i3 Y/ t% C                  ^
8 T1 V/ u6 O) D, ]5 j3 e" [1 CRuntime error in 'FundamentalUnit': Field must have positive discriminant

! s4 W/ t4 g, M1 X+ w, F
>> FundamentalUnit(M);
                  ^
$ k0 v+ z+ l( N* M  X2 [" K7 vRuntime error in 'FundamentalUnit': Field must have positive discriminant

20
" I0 O. E* _. ~6 M# C2 X
9 W. e4 t  r3 e6 _" R>> Name(M, -5);
9 q  R( [* ^& ]) T       ^
6 N' W" g; s1 [$ p7 VRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]
" K' G% F, r7 ^; j" Q+ M1 n* J+ |
1
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
" J  M" l9 n7 W5 P1 w/ tDefined on 1 generator
1 a9 w) C! q5 p3 h9 YRelations:
    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
Defined on 1 generator
$ c5 ~  k! [4 W' b7 m% d/ LRelations:
: ^; _# _+ G, y$ j7 F3 {    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
7 N. r/ k- t) ^% cDefined on 1 generator
Relations:
& P4 D+ S& c$ }$ ?! \. O    2*$.1 = 0 to Set of ideals of M
2
' k' g0 s- V- `+ `/ y0 E, I2
; }) ]0 M5 y, ?; L  q0 z  A& u, [Abelian Group isomorphic to Z/2
Defined on 1 generator
  W& V' O9 k8 T; G: q0 KRelations:
5 G2 L* D4 a" |% H. K    2*$.1 = 0
9 `3 U4 U1 ?9 E7 B1 OMapping from: Abelian Group isomorphic to Z/2
- c! x  Q, e! M7 R' o$ g4 v5 Z( aDefined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
5 e1 o6 Q$ B/ j. R, g2
) w: S. q4 b# _, M% WAbelian Group isomorphic to Z/2
& @" o" r5 ?6 ]* p& Y. YDefined on 1 generator
! m8 K9 X$ k! @! |6 ORelations:
    2*$.1 = 0
1 }9 t! f9 |+ a( J3 }Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
8 f' t* F8 E( A; o9 c% Iinverse]
) \- _& x3 d5 y3 s2 U  b/ D3 wAbelian Group isomorphic to Z/2
5 e: P5 _- T4 q) l! y9 u- EDefined on 1 generator
  z9 d8 A; q- m( t( c5 mRelations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
4 l" z9 J2 N3 q1 W( V; J4 F9 [& YDefined on 1 generator
Relations:
$ m2 j. \- L1 k    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
* m" f3 l9 Q/ F" S, a6 \4 Minverse]
false
0 `1 h+ d. a& i& `* ]false
7 M# P$ ?; S3 F. _: i% Z7 E* i8 h==============
( n4 A. x6 H9 q
. x' e9 k; {, f7 Y
Q5:=QuadraticField(-50) ;
Q5;

$ b2 x+ H" X' V- G/ H2 e4 ?Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
9 ^2 w2 m  ?# ~& p6 [$ dM:=MaximalOrder(Q5) ;
: Y8 n) |) u2 p3 {2 n* V3 {2 W5 P( BM;
* G1 t: F, N5 S  v! z  @. uNumberField(M);
' h5 G0 ^1 ~0 S- w, h; eS1:=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 I8 e4 q5 x$ p! E9 s6 w3 @IsQuadratic(Q5);
+ c0 o; l# q/ S# `! OIsQuadratic(S1);
% d3 j* s3 }( DIsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
6 W  U/ y, M, Q; y2 NConductor(Q5) ;

# w/ }7 {/ R' S* Z/ t0 {Name(M, -50);
; z: F! n$ {7 L9 y1 \Conductor(M);
ClassGroup(Q5) ;
$ P3 G. E- ~% T" D; JClassGroup(M);
9 {: Z& u0 m5 q9 w8 H3 C3 j5 F3 d& LClassNumber(Q5) ;
ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
/ G: O" a# Y" w! Z3 K
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
6 t& r9 @& b- }( o; J5 K/ |+ z2 X: cNormEquation(M, -50) ;
$ i& u* ]. ]6 ?- y% b
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
6 r1 _& [9 v/ p9 a0 M6 P/ k, _Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
/ y1 K1 J. a1 @6 {8 [Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
+ b6 P5 R. H$ y: Y4 btrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
' N4 K/ n; u" _0 p2 w2 q" H' rtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
; `4 ]. l6 B& ]# z1 t% Ttrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
) L2 \+ V4 q3 S0 b[
    <w - 5*Q5.1, 1>,
    <w + 5*Q5.1, 1>
9 h6 ?4 v9 M7 u# l]
( M5 B2 G1 n8 b5 b1 z) L-8
) t- u8 I: a, K4 t* Y* ?; \1 n- X
>> FundamentalUnit(Q5) ;
3 r' h+ F7 [3 n- S+ D                  ^
! w! a* O" h+ Y" J8 y; r! aRuntime error in 'FundamentalUnit': Field must have positive discriminant
& Y* ~1 N2 x8 ?+ E( z7 r7 v5 y

  N+ O. W# M* j1 Q0 o& p' k( ]7 s7 [>> FundamentalUnit(M);
: ~4 r9 O! m: @5 \/ P5 v" m. M                  ^
+ ?' T3 _2 X  y& Q/ FRuntime error in 'FundamentalUnit': Field must have positive discriminant
% h$ E9 F8 U5 @  Y7 E' i2 x8 r* Z4 B% I
! g6 l  \7 p8 @1 W6 W8

  ], I* S5 J3 B# V4 s>> Name(M, -50);
       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

  p: c5 I% Y' J) P9 y4 `1
Abelian Group of order 1
9 t1 ~, f$ ^# e8 b2 W4 kMapping 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
) C8 G: e! \! x3 w4 o1
$ g. T/ r% Y1 p3 R1
( Y# s1 g$ u3 Q" U3 r' L. VAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
9 m2 L/ A" |! w' F5 I/ K# \1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
2 Y3 M9 j* f2 v. a/ ?* o-8 given by a rule [no inverse]
false
false
7 R6 z5 b' z; X0 Z- C

lilianjie

看看-1.-3的两种：

Q5:=QuadraticField(-1) ;
( m; H6 M7 ]* K" qQ5;
  J/ ?1 x, z. y# D6 f' }8 Y3 z* a% f
Q<w> :=PolynomialRing(Q5);Q;
( W! J8 O3 I8 U2 @EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
8 V$ y# M! q2 o  vM;
NumberField(M);
6 K: d$ N1 Z! X6 r' P' W+ KS1:=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( w& U+ P' w7 AIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
7 [5 `5 b) }, D+ w9 i. Q% @Factorization(w^2+1);  
8 T8 z0 Q6 G) l( m; |1 N8 JDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;

( t* {: |1 y$ O2 d* o7 w# vName(M, -1);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
3 Y. A$ _$ B, g8 j5 m4 dPicardGroup(M) ;
- I0 ]7 _' L4 [- nPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
; V8 u  ]! Z- d  T& s7 `+ y. XQuadraticClassGroupTwoPart(M);
1 J' ]  u8 [; cNormEquation(Q5, -1) ;
  w7 v  ^4 b, ]9 h0 NNormEquation(M, -1) ;
- @, D' }' O+ I4 L8 e8 G6 t7 r7 ?
/ q1 w5 x& n: P3 @, L! iQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Univariate Polynomial Ring in w over Q5
4 g$ Y6 K1 r3 Y0 d: s0 b" |Equation Order of conductor 1 in Q5
0 \# s, G" o) d( U  SMaximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
; b0 f1 f$ s0 ?& a& ^9 a  E; z' Ztrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
7 c; X& \- Q2 S1 r3 c6 {6 A0 Ytrue Maximal Equation Order of Q5
) @8 W; G) w: Z- i7 {+ e' xtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
4 [0 T4 b7 W& Q5 L( jtrue Order of conductor 1 in Q5
[
7 Y/ J* K, a1 A" y    <w - Q5.1, 1>,
7 J! r6 o& P5 ^& O    <w + Q5.1, 1>
]
  I, ~" N5 s- l  |9 T! t-4
+ {+ m& {6 m0 ?  A# I- t
" m1 R9 T* Q" I& i: b1 ]; F>> FundamentalUnit(Q5) ;
+ o9 C0 W) p9 ]9 A; G1 R6 ~                  ^
* V% v# A: n0 c! A+ y' k" sRuntime error in 'FundamentalUnit': Field must have positive discriminant
) @$ l1 H' e4 E7 J
7 p; t- _7 ]' w9 z7 s* O
& B3 e* U, U# l2 r+ {; m5 L>> FundamentalUnit(M);
                  ^
6 Q  R' C& y" b! k5 a+ ]Runtime error in 'FundamentalUnit': Field must have positive discriminant

6 J" K( l2 x- p) w0 R- k7 C+ w; ?4

>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
2 I9 y8 x; B1 K) C8 p
! ?( q8 v+ c% N& m4 l1
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 x6 ?5 d6 G, i- C" t1
0 n/ G+ C, D# \4 e0 ]0 o, B1
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 w9 E; P! i6 l7 T5 @) I  O8 C& y1
9 X9 M) @* u  f/ i3 y" |Abelian Group of order 1
8 [5 A3 ~( q0 jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 S# T4 x* R1 W: w" p. l-4 given by a rule [no inverse]
Abelian Group of order 1
* u/ q9 h" U8 F3 O9 {. YMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
( a3 a( [" \3 n  T5 K4 q1 Efalse
2 S/ P3 U. S. I7 ]( t9 x$ dfalse
===============

Q5:=QuadraticField(-3) ;
Q5;
+ Z+ F6 `+ _% r" G9 [
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
* k$ `. ^' a  |) v+ A+ L3 w9 O2 m% hNumberField(M);
9 l* j. K, |* Z0 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);
IsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
9 H4 ]. ~( A- @) UFactorization(w^2+3);  
/ l. J: }* s% EDiscriminant(Q5) ;
4 w9 `  `( {: l6 g, _FundamentalUnit(Q5) ;
2 E" A9 H3 F1 I9 e; WFundamentalUnit(M);
Conductor(Q5) ;

1 `( Z* o' y9 R, AName(M, -3);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
) l! D9 n1 }7 \$ iClassNumber(Q5) ;
ClassNumber(M) ;
0 z$ ?5 N. o2 O3 k/ j, i! B) J, ^PicardGroup(M) ;
3 o8 q1 Z7 l( r8 o* qPicardNumber(M) ;

% T/ h( D8 p* O0 m; SQuadraticClassGroupTwoPart(Q5);
* g& [9 l5 J% z3 b7 Z$ `7 WQuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
NormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
0 @1 _4 v( P! U8 p( yEquation Order of conductor 2 in Q5
5 U( F; M8 F3 w% J) YMaximal Order of Q5
  r8 u8 h0 o% Q. F% c4 P3 A( ]Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
/ J" m, `# O% i- k" q2 t% D/ Y, V( v* aOrder of conductor 625888888 in Q5
6 M& h3 h- B( X5 r( C7 Q/ M# Rtrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
0 N& }; ?4 H' g- g$ o5 {* Atrue Maximal Order of Q5
true Order of conductor 16 in Q5
2 D: D* C9 B* x  v  L0 @. ftrue Order of conductor 625 in Q5
true Order of conductor 391736900121876544 in Q5
[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
2 z: w# S  x4 S& p4 J]
: h; l9 [9 g9 ~( P  d-3

3 x7 O; M" f1 K>> FundamentalUnit(Q5) ;
! v/ B' q% v% d" X. p, w; w" ^* E5 S                  ^
+ T  g  v  F9 E8 yRuntime error in 'FundamentalUnit': Field must have positive discriminant
) p4 D: i, }5 |
8 H+ F8 x2 S4 T9 [- C" z
$ T" {7 y$ m( O2 f6 F>> FundamentalUnit(M);
                  ^
. `$ |  j5 T3 Y) \& ~& _Runtime error in 'FundamentalUnit': Field must have positive discriminant
9 s  |! ?7 [9 f$ Y1 ~. \9 n' p
% h" I0 e) o, M$ e3
! \5 C  S- o6 m5 X
' I9 G" R* i2 a( j>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
2 A# F2 q. ]: M6 c! c9 M6 q
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
8 Y/ ]" P4 R. v& b2 Y& aMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
/ }" L# A+ R; e8 M1 K# rAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
* e6 y! F+ t, W7 g6 f0 iinverse]
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
9 ~1 ^( `+ j" O5 w5 n-3 given by a rule [no inverse]
Abelian Group of order 1
& K9 k1 N3 c% Q* M/ Y2 J% vMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
5 ?) N: w% T' L4 a: E+ }  o-3 given by a rule [no inverse]
false
( f+ Q# _  }# ufalse

孤寂冷逍遥

lilianjie

Dirichlet character
! u' q; @5 ?# T' t* [& EDirichlet class number formula
% E$ ]2 S* X& L0 e9 r
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

6 j! ^5 Q: H5 ~9 c/ X! G/ j-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
% f5 z! p1 ], K1 [" m+ _8 R4 x! G
-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
: O2 B! _* h% Fh=-6/（2*3）*Σ[1*1+（2*（-1）]=1

8 i' E- Q5 |: Z' }-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，
: z' T! M! x! m5 W7 B4 x
/ M& _6 ?4 k7 w9 H
* _) ]2 K$ T7 a  y6 w
) F. q6 E. @: O+ Wh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2


" a1 x6 I0 g, R- l6 ]% B% E
$ J; \: l8 e/ J/ p" u-50时  个单位根                          N=200
) M: C& g+ H/ q

lilianjie1

Dirichlet character

lilianjie1

pell   equation

lilianjie

% o% `; c% q) `4 c. p( S2 vF := QuadraticField(NextPrime(5));
9 `& W) m% I+ m! s. U- k+ D- k5 P
KK := QuadraticField(7);KK;
+ i" @4 B- |" w& q' U: ^K:=MaximalOrder(KK);
) m4 g* p  l" V+ k9 c. OConductor(KK);
8 p6 {* Z6 X4 N. A$ MClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
$ N+ {+ |' c  |4 v2 J) PNormEquation(F, 7);
- _4 m& |6 T% y2 W" j- vA:=K!7;A;
2 k# Y1 y8 k/ ?* l  i8 X) Y" \! n6 @B:=K!14;B;
! |& H3 E3 F, t1 XDiscriminant(KK)
7 O' d: h3 P- u1 ?
Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
2 Q" N" k) D3 v# E# @2 v28
Abelian Group of order 1
* W( O& z' p' H% l7 Q# F. B  nMapping from: Abelian Group of order 1 to Set of ideals of K
Abelian Group isomorphic to Z/2
6 J% d4 n7 g# I& C# P; Z& k8 NDefined on 1 generator
Relations:
0 ^; O5 K. ?/ V    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
7 M; I! C8 K$ c! R/ n6 M5 y8 CRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
: o5 Z: i0 t7 Z* \8 o3 [, L' @false
7
' E4 }& P# _* ?9 X4 A4 a6 e14
28

lilianjie

  ~- t8 g2 x  v/ @, W
分圆域：
C:=CyclotomicField(5);C;
6 {$ n4 I5 y& Q; [5 j1 o: l3 wCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
( m/ f2 V8 e! s6 o$ Y, j- BCyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
/ q) m; V# ]4 fMinimalField(CC!8) ;
6 a' b  i4 y0 R* t: H7 L! ]2 KMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
- G, z4 A2 |2 n, [) _* KConductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;
3 S* ^. p: r% Y
: `+ I- }3 D, [5 dCyclotomic Field of order 5 and degree 4
7 X2 j5 L# U" z. r$.1^4 + $.1^3 + $.1^2 + $.1 + 1
& Z: e+ F% i3 mCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
( J' I7 I/ l. d, f- F: vCyclotomic Field of order 7 and degree 6
* U& M, s6 }6 r$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
9 Z' s- _8 A+ @, ]Rational Field
& @5 k* ~: |* E: c- FRational Field
$ _3 x0 {$ ]3 m+ d; ?Rational Field
6 c8 j. I9 D9 c7 B: k3 azeta_11
, h' F. {( l  ^8 u% [zeta_111
( H# q8 ^1 x) H4 O% y7 n# b123
7
7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
    (1, 2)(3, 5)(4, 6)
. s+ W& m3 x9 n6 n5 p3 ]+ J; G0 j    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
7 B- w7 @6 e1 \CC
- {5 x- W6 `& l4 bComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
" J4 n7 q7 D6 ]* W5 CDegree 6, Order 2 * 3 and
, s# F3 r7 w% m2 d( m4 Q' M+ rMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

5 O" R. [8 }# Z5 r9 S+ r+ I

lilianjie 发表于 2012-1-9 20:44
- k# z% M6 S" r$ k' W& s  k6 |分圆域：
C:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

* p! u5 {) g  p5 B& G
分圆域：
分圆域：123

; R" z# }/ P5 A) ]0 Y' t3 [  f- HR.<x> = Q[]
$ b- N" b/ ~4 q6 {% p" H( oF8 = factor(x^8 - 1)
  s( y4 X, n, |$ k; t! {0 bF8
/ d. ^8 v4 ?$ u) `
(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;
' p6 W$ T' I8 E" K3 K2 rC:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;
# q, p4 }' P# s) t
F := QuadraticField(8);
4 t& a2 D/ G0 `- M( lF;
. S: Y) [2 |2 ]* ^/ f; VD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
8 I. J# k6 h! x. C3 b: Q8 L2 YCyclotomic Field of order 8 and degree 4
$.1^4 + 1
' M1 c9 H* }( c" \Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
) `4 R6 k, H5 h' S[
    <$.1^4 + 1, 1>
% g- K7 q/ J3 Y0 d]
, m/ f' q! T2 E4 r
# t7 J8 p3 ^/ G! c  M  zR.<x> = QQ[]
F6 = factor(x^6 - 1)
7 U& ]0 G1 }/ u# ?( PF6
! t' c8 [& `; n# q8 O1 J5 X
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

# s/ N" K$ j0 n4 p$ \0 AQ<x> := QuadraticField(6);Q;
1 O) k( D' x/ H0 kC:=CyclotomicField(6);C;
$ z. z( J$ t. a6 ^! O, UFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
3 H) N) I- {, W& B; EQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]
! Q- a" q: w5 K- g  M0 B! b
8 l7 g* {! A7 L. oR.<x> = QQ[]
F5 = factor(x^10 - 1)
( T9 Q8 R& O7 \7 d  ZF5
2 c( W( `! x/ M5 l* \* K(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;
2 q2 A$ R% c5 uC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
- f7 N, O  L, mF;
) W( x$ o8 o% [7 cD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
1 N! v7 Z) \( n# n" |6 R# q* t, C/ OCyclotomic Field of order 10 and degree 4
: f1 K9 w+ @# P) Q! b$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
2 A; L6 a0 e. l# p+ D  Y[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie1

分圆域：123