一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
> n;

7 B; Z( T* R7 |0 \> n:Hex;                           转16
$ v. v$ k7 N# N( _' V  rIntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
: @- X. o( q$ B: \0 E9 I  |' GIntegerToString(n, 36);         转36IntegerToString(n) ;
( p+ |7 X% c2 h0 b9 hIntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
: D& w8 G( ]- ]2 n9 z+ Q" H/ Y, ]Representative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
/ b" n0 T: F% |/ \& e" y7 f3 I
/ v/ i+ C1 h  n5 S  hm := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
& g/ F8 c- m6 Ekk eq k;

k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
- C2 p9 j$ N  b, s> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
1 G1 H9 ^6 u; S3 U( ]> kk;
0 h' [/ {* Z$ o  ~' ?& |kk eq k;

$ S  {# g  B( O2 l* v' e' e% v2 u1 mEltseq(kk) ;Eltseq(-1/14);
. Z9 n! I  A4 R7 o  U4 Z* Y
/ A* [0 \: d" T. M3 Y

4 g3 f( ]  W+ d* P( t; E7 z
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
n eq k;
; X- F& {) Q& f8 skk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
* z/ {& n+ j3 Q! m
Eltseq(kk) ;Eltseq(-1/14);


3 @% ?" O; ]" @; v% P% E' n
0 A# V4 k: f+ s5 }( U
+ U9 s) g6 R$ ]( y+ O

& z6 e- D8 a% Z, `# l6 n# u

/ K2 R5 W# ~7 F' \. U

, t" ^  H/ x- y( G) x) e/ c- D5 b2 ]=============



2 B. P3 h" Z. F
Integer Ring
" I$ i/ f" F: y+ R3 c. W-1666666666234567890
-0x17213080A7E55CD2
4 f2 i0 U* m/ I0 _-1011100100001001100001000000010100111111001010101110011010010
2 C& c- S" R2 e-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
0
1
0 [1 _9 D2 W0 Z* s) ^0
. k& `2 Y2 X% l& D[ -1666666666234567890 ]
! e3 F$ Y$ m  e6 X  D' J  c6 Q[ -1666666666234567890 ]
1
13
1
; u- q8 f5 S7 J$ ~& z( M
-1666666666234567890
true
( T4 M8 S+ a; p8 e-1666666666234567890
% m9 b( I# D+ y4 @$ ?true
-1666666666234567890
5 _+ _8 a% y7 u& ztrue
) z6 u5 U& v; M; p) W5 `-1666666666234567890
true
[ -1666666666234567890 ]
% F8 p* ]# M: b) A" p[ -1/14 ]
9 D3 N* |4 ~, y! B- z* Z3 f/ d
! C$ K6 i) B& Y7 t3 z


+ m9 r( b  T0 s! J# o' G0 P" v
/ A( @3 w% X0 f1 _

-1666666666234567890
. i2 H( r$ r) r1 W0 J6 u- J-1666666666234567890
! k# ]* G) s9 {5 _2 o. I. k1 L0 ?% m7 Jtrue
3 Q8 b7 X( X; L& X% x-1666666666234567890
$ m2 I! @7 {8 q$ i6 ~2 v. }6 Qtrue
-1666666666234567890
1 u- z& M; k1 d  ~. k3 Atrue
1 I5 |0 t) N# e/ g: E* @1 ^* z- \-1666666666234567890
true
8 S) \, f" H) a& v[ -1666666666234567890 ]
[ -1/14 ]

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
s:=0x12345678111;ss;

sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
1 a; I  ^2 _+ Y3 U+ p: FFactorizationToInteger(sss);
* N8 v5 ?/ n/ Y( g- ?% \7 dFactorisationToInteger(sss1) ;
# O, d. Z$ [8 P; s! XFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
5 a: p. q6 W6 ^( ]* O) F( Essss:=Intseq(ss, 17);ssss;
$ {" o0 }- q% K$ S) h/ \SequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
6 |. ~; `4 ~0 T2 aSequenceToInteger(ssss1, 17);转成2和17进制
/ ]( K  Q1 l+ S$ ^" d
6 f- x' m$ E# u& G+ l12345678111
" i; O$ B8 h' j8 F12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
1250999894289
[ 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1,
# a% P0 F' J/ f; U1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
$ J  L% u, ~2 X5 z6 I1250999894289

lilianjie

Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
1 M" C7 X  O; Y8 l. S2 |0 ]' P1 Gn1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;


K:=Z!n1+Z!n2;K;

7 H% w- Z# R5 K- ^9 W3 A" f- IIsField(Z);      是域吗Characteristic(Z);环特征
& f" Y8 y/ o: W' q; B! Y' y, NIsFinite(Z);有限环吗
IsCommutative(Z);可换吗
IsOrdered(Z);有序吗-------应有不过这函数没有这功能IsEuclideanDomain(Z);欧整环吗------欧环还有非整的。。。。
2 o' A4 W- e( D% h; I9 z: q5 hIsPID(Z) ;主理想整环吗
0 O$ S* Y6 ^' B1 z0 ^, \; c
0 F  {, F7 `5 B4 mIsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
/ [6 Q% M) c6 iIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
/ G6 }- p; X+ ]0 \: o! |MultiplicativeGroup(Z);乘群
Category(Z) ;范畴Parent(Z) ;父环
PrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
) Q% T: ]$ X8 h# s1 m4 EAdditiveGroup(Z) ;加群-----就第行一特点ClassGroup(Z) ;类群----------只有Z才有------难懂理想类群更难懂
" i( Q$ r5 C5 T4 K2 E% q) H9 E
4 }$ P% X* l5 b7 zZZ:=IntegerRing() ;ZZ;
6 K4 M0 [, r; t+ W0 a# pClassGroup(ZZ) ;

. Z3 [. x& ]: y8 I3 B2 @5 N% [===========
- [! O. d: G8 a  S8 X  G
Residue class ring of integers modulo 5
$ w7 U2 {5 g; @  Z1666666666234567890
1
4 m8 ], p4 K; h! ^' M) m$ v1
2
true
5
true 5
true
false
true
true
true
/ s' A- w5 B# @. Z, |% W- atrue
true
true
true
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
# j3 K; q6 ?# K# r6 ^6 F' T& e4 e; K2 VAbelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
6 _6 ]2 x  E+ E( LRngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
8 b4 v# T$ i6 }* J6 QAbelian Group isomorphic to Z/5
Defined on 1 generator
! N6 }& d# B: t+ U  \Relations:
/ P9 p$ \5 y) P6 }* u+ f5 w    5*$.1 = 0

& C" J# n+ T+ V" Q, Q>> ClassGroup(Z) ;
" X. ^/ Y  Z% `. i) L# s             ^
* q1 S6 r  z, H+ Z6 c0 p+ }6 Y/ }/ vRuntime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

Integer Ring
3 Y- e; Q9 a* a) H7 h* f5 [/ `- K/ |Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
2 {) l# U  B; n5 z' T8 P UnitGroup(Z);
5 s4 z  B4 I- v3 ]" UMultiplicativeGroup(Z);
Category(Z) ;
- m! T" w/ c& Z2 APrimeRing(Z);
AdditiveGroup(Z) ;

$ C" p& A% ?- z5 MZ:=IntegerRing(13) ;Z;   
, Q1 ?' i5 g  X/ |' h4 ?/ l UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
8 L- Y0 R+ `0 L% kPrimeRing(Z);
- t, @" u# S1 X) O0 u0 Z1 Q: O" {% |AdditiveGroup(Z) ;

7 g5 F; H! y7 \9 m, D' B0 |
; c6 I$ `; f/ H4 b+ J0 a3 t
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
: P9 j( b+ w2 _+ t0 i& NRelations:
    2*$.1 = 0
    2*$.2 = 0
3 @. S5 I) L! aAbelian Group isomorphic to Z/2 + Z/2非素数环的乘群同构两个小群的直积（1*11    5*7）Defined on 2 generators
Relations:    2*$.1 = 0
" n: d% P  e% d* O' V( }! f    2*$.2 = 0
" l' {; b, v# u9 Z  ZRngIntRes
. c5 j6 k7 J, S/ hResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
) ~' n  Z! R# q7 NDefined on 1 generator
Relations:
    12*$.1 = 0
Residue class ring of integers modulo 13
$ g5 a" K% Q$ P; H& {Abelian Group isomorphic to Z/12
Defined on 1 generator
2 D( |/ |& Y  X/ @( `8 Q+ v, w! NRelations:
* Y4 ?* z+ D% ^* w) O  s    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
7 u- C% ^0 w6 Q# @4 @& z" cRelations:
    12*$.1 = 0
RngIntRes
Residue class ring of integers modulo 13
! t2 U! v; s" k+ q" H& qAbelian Group isomorphic to Z/13
Defined on 1 generator
0 G9 y; W2 d: l5 M* ARelations:
( J# M! c/ G2 S/ N# n2 P    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   
: v9 Q1 b: ]- c# ~' a
R:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   

# K& a$ M9 v! \6 A2 B, [
PrimeRing(R) ;
" ]- d" p% _* f. RCentre(R) ;

Characteristic(R) ;
8 @+ V, O& f1 L( a4 |# g# R ;阶----元素数
IsPID(R) ;非素数不是整环不是极大理想整环，但都有极大理想公因子IsDomain(R) ;
Has**(R) ;

% z  ]8 ^' Q% R' X4 ^) @IsPID(S) ;
IsDomain(S) ;
$ g( V6 }% `! K: W5 P- wHas**(S) ;
: k( V# `# Y0 \( K' mR eq S ;
R ne S ;

& U0 Z1 h4 t7 ?1 A( u6 _* aParent(R!123) arent(S!123) ;
Category(R!234) ;Category(S!234) ;

3 z2 a* Q4 ^; S, L* F9 O0 [a:=Random(R) ;a;b:=Random(S) ;b;
! _( I- e. O2 P- K% L; ^Representative(R) ;
( _% l# p( {' Q+ MRepresentative(S) ;
  M1 I( h: |/ `: {6 n' R
(R!a) in R ;
% \- W2 u/ T' O5 h0 g! U# _(S!b) notin S ;
IsUnit(a) ;                是单位吗
# ^, D  k, f. P  L$ p0 _4 qIsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
1 X8 {. N. Q0 o6 a% pIsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

9 r3 [3 x, t9 v# aZ!a gt Z!b ;
1 j3 }* O: X" gZ!a ge Z!b ;
  G1 v7 |: T& A  a( q8 A  EZ!a lt Z!b ;
Z!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
Minimum(Z) ;

Maximum(S) ;
Minimum(Z!a, Z!b) ;
  V/ ?% g5 Y2 GMinimum(R) ;
% J" }3 D: c9 g2 p  z
8 m2 W$ \! {6 o3 r% L

Integer Ring
* _$ y& f9 U- F! n( }6 jResidue class ring of integers modulo 12
3 W- a" G+ O& ?/ J% O( n* tResidue class ring of integers modulo 13
- T% X$ k* m8 l' L$ E; o, _/ W' AResidue class ring of integers modulo 12
/ Y" [: N1 l1 W. v- I# j8 rResidue class ring of integers modulo 12
12
9 U7 G- Q+ S* u: x12
; o3 z, `9 ^- K6 mfalse
) q, ]' C5 B$ J$ F" Ofalse
* m4 Z3 \" V% _; O$ dtrue
8 @$ N' R8 |. T; E9 etrue
' q) \) g' k$ Z! i* B4 z( O& Ztrue
true
" I7 d+ J+ M( S' W+ j6 i" D7 Afalse
0 p7 U- o4 @8 S) dtrue
1 G. ^- Q/ M1 ?% f8 OResidue class ring of integers modulo 12
, N) m/ W. H2 B1 n  k! F2 @Residue class ring of integers modulo 13
RngIntResElt
. w2 N4 [: F" u6 NRngIntResElt
* L4 j% ]3 b/ w/ G5 j* R; l9
, M8 `9 Y9 b  U8 O2 [6 |12
& X$ ?0 w! \' Q- D0
' u+ X9 t6 P, w0
true
6 G( h2 G$ O. C8 Ifalse
false
3 A! m( H+ T" \; U" Rtrue
8 f! z7 V4 \" d( Q/ `9 Afalse
true
: J9 p5 E* J+ s0 B6 o0 T$ J  Afalse
) |1 _+ j4 g) T4 Dfalse
false
false
true
true
12
1
+ j( I1 @' s1 ^9 @
+ O4 R5 x% d  Y( ~, c/ h>> Maximum(S) ;
1 n0 |6 ]" j" ~3 T+ L$ F1 E- e3 X          ^
Runtime error in 'Maximum': Bad argument types
Argument types given: RngIntRes

9
4 p8 z6 U) h8 i# A5 O1 v
>> Minimum(R) ;
          ^
4 b5 u1 g9 w( L* c% y/ M9 f: B3 DRuntime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

lilianjie1

- a% A. J) L9 LZ:=IntegerRing() ;Z;   
5 s% {- B* a. z( O1 |$ kI12:=ideal< Z | 12 >;
1 u+ N/ V( V) u0 q- \7 ~I12;
ZZ:=IntegerRing(15) ;ZZ;   
. y( e- L% z* O* ~& eIZZ15:=ideal< Z | 15 >;
  F/ c. s  S/ Z3 E8 S. g0 zIZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
, v8 q- |; \% \6 ?ZZZ:=IntegerRing(5) ;ZZ;   
" e1 a$ ]3 ]; w2 m: F) y: D6 p$ U% GIZZZ5:=ideal< Z | 5 >;
IZZZ5;

I12 *  IZZ15;            理想和/积/并/交，
理想和是理想对应两（可多个）元素加，
& U9 o4 r; c. c理想积是两理想（可多个）对应元素积，
理想并就两（可多个）理想元素并，就不一定还是理想，
理想交是理想（可多个）元素交，理想交一定还是理想，

  V0 g: `% `6 v, K4 H2 E" z理想积是理想交的真子集，极大理想交是理想------J根
理想商就理想间同态：是必须能整除
I12 +  IZZ15;
I12 meet  IZZ15;

I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
& a" |* r  `1 h5 iI12 / IZZZ5;
7 y9 {! G- ?/ ?IZZZ5/ I12 ;
. D& v# {0 B( O, KZ * IZZZ5;
I12 + IZZZ5;
, P6 [& c; B8 H& t5 XIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;

I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
/ Y8 K" c! P/ z% @" wIZZZ5 subset IZZ15;
Integer Ring
- Y* U2 G% D2 q) [3 b  OIdeal of Integer Ring generated by 12
: e4 X" U4 |9 ]4 l1 |Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
false
' S; P9 t' E- D% U' X  l. [4 D6 xResidue class ring of integers modulo 12
Residue class ring of integers modulo 15
" }2 d- ~- r, L+ E! [Ideal of Integer Ring generated by 5
& Y# N6 K: A9 g) P7 b3 yIdeal of Integer Ring generated by 180
Ideal of Integer Ring generated by 3
# ]) ]( J- e. j& R* pIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
# q- s8 }( d; Z3 E$ xInteger Ring
8 x0 i4 j5 |& o! yIdeal of Integer Ring generated by 60

>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.


3 |( K9 G. i7 u0 _6 Y* c>> IZZZ5/ I12 ;
        ^
5 T  v( Y% }1 ]7 ARuntime error in '/': Argument 2 must divide argument 1.

& Z* r5 j. M- T3 BIdeal of Integer Ring generated by 5
9 V/ F3 Z+ t" ~" k. \( [3 O7 eInteger Ring
Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
, b0 G1 a1 v$ BIdeal of Integer Ring generated by 5
1 @* n5 G' L! s; ?  j" tIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
: k0 b$ n" M7 t% E, o) s/ lIdeal of Integer Ring generated by 3
. j6 p6 |+ f4 |: OMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
  o9 {& A5 Y5 [- a5 _* X8 A& Q0 g
4 E( y+ ]  B; R: w5 Rfalse
$ g6 S3 D' a9 {8 X' G& otrue
true
7 w! v$ d& G4 b+ N) `4 E: o9 Nfalse

lilianjie1

Z:=IntegerRing() ;Z;   
7 N! R* Y; r6 v- K0 TI12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;   
- X- Z* Y  ^9 G( K- S& qIZZ15:=ideal< ZZ | 31 >;
4 D& K' I. I( h0 C4 @1 I6 z# ?IZZ15;
ResidueClassField(I12);
ResidueClassField(IZZ15);环和极大理想的商构成域---剩余类域，剩余类环中的素数都是极大理想
loc< Z | 19> ;
loc< Z | 17> ;
+ C, m% E9 o6 ]loc< Z | 131> ;局部化：一个素理想到原环元素的映射
ext< Z | > ;超越扩张到一元多项式
ext< ZZ | > ;
7 ~' P; n* g) \/ N* Y, o
ext< Z, 2 | > ;超越扩张到多元多项式
  |! D$ t0 b2 E8 D
ext< Z, 3 | > Completion(Z, I12) ;
; @, F, x3 U& g
% X- w" D3 ]4 v- B; W
& C0 h4 _; ], m. w$ K& A5 x. n* _" |comp<Z |I12  >;
    素理想零理想完备化，和P进环联系起来
1 g$ c: @8 \7 bCompletion(Z, 0) ;
5 J8 ~3 p  [) e; Z) [. k, jcomp<Z |0  >;
% Q$ d) n  w- ]/ i$ f& G8 t4 Q
Integer Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
9 G7 |, ]3 _7 w( J; T2 QFinite field of size 13
% D6 G" k  X  z  l8 d  V& ~) _Mapping from: RngInt: Z to GF(13)
modulo 13

9 I. q4 p7 q; {/ d* @- m% r>> ResidueClassField(IZZ15);
( H& j7 C0 R- v2 N                    ^
Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes

Valuation ring of Rational Field with generator 19
) i4 |8 t& `' h/ I4 Z* DMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
# _& p# i2 H& J" l  dValuation ring of Rational Field with generator 17
0 P! h' R# {' BMapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
/ u5 M) Y1 N& Y2 R* `2 U) TUnivariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)

3 ?( p; D" M* T% l: q0 S0 u; ^>> ext< Z, 2 | > ;
5 \+ k" g' K# `  d' X      ^
Runtime error: This constructer is no longer supported


>> ext< Z, 3 | >
      ^
  s2 @9 c  o& P' yRuntime error: This constructer is no longer supported

13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
3 M% }9 m4 \! X* ]2 J+ S
/ r6 q. {: ?! O2 {  ^4 SCompletion(
3 |% M1 ]5 m9 D' Z3 y7 P# a    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0
3 b% h6 z1 g+ E( ]) d. v, i7 I