一些初等函数：

lilianjie

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

8 ~) C7 I  n3 I) J4 R> n:Hex;                           转16
IntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
8 ]" ]4 u/ c1 w( z0 b! f9 ]+ GIntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
7 g+ I" `& L& FIdentity(Z);                  
, n! c& r3 n/ u2 E- ^7 HRepresentative(Z);         环代表元
8 @: p2 s+ k5 U' u  EEltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);

m := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
: V& ^: x' F0 m6 T& Z( H  Xk := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
+ U& d7 L0 L' t% i5 n( J0 Q- j> k;
4 {& W; _) M# N5 in eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
) ]6 r4 ~2 Z) Mkk eq k;

" C0 u, |- @+ E  pk := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
- M+ E. u  t' l> kk;
2 g. f, N' ~2 ]8 J3 nkk eq k;
' U% E5 y' U' H. w* [9 n; x
Eltseq(kk) ;Eltseq(-1/14);
' H" {+ D! g. J7 V2 P5 y, q
% s9 b& g1 D2 m5 a) Z' n5 U9 ^
: @. Y1 b8 s  r3 d% v) \
& i8 ?, @$ `) p8 K, w+ R
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
6 `- U' H; D9 I  R) {8 U  y1 f> k;
n eq k;
3 U0 M" k0 j, I7 s0 Q  K- C  ^kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
' x& p6 q8 M% K6 N6 y- I+ C1 Z3 s> kk;
kk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
/ M5 k: n9 i# D- M/ Z' wn eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

1 r5 [( M7 \* n' O1 LEltseq(kk) ;Eltseq(-1/14);



# y& X& j  L$ P& G+ J


. P5 r2 N# f3 _5 x

5 }2 |- |+ ]  \% r* \" T8 r+ b
3 u/ V( b( G2 u" k
=============
: u4 Z8 t; l- N
# O+ F4 J5 T2 [) j; w# T

* y# r/ u4 f, ~3 l: c3 \, q
Integer Ring
! B& D% f) d% \6 C. R! B4 z% I, N-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
4 i5 m) T2 Y3 x. K2 b5 ?-1666666666234567890
# [# D* j: p4 ^' g# r$ [" B-17213080A7E55CD2
) k8 Q7 }( \! b- J* ^-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
- B, w0 a6 {, O0 z# o2 S# `! Z0
1 J! s; h1 k" j( \1
# V( {+ _* d) n$ U0 T1 Z2 ?) X5 W4 O0
[ -1666666666234567890 ]
, y6 Q9 _  e9 U* j[ -1666666666234567890 ]
0 v& F5 \$ d/ y; F; r# j8 G# S1
13
1

% k$ @+ H3 h- P; L3 a# G-1666666666234567890
true
-1666666666234567890
  E( {) p8 p! B- l0 H+ R/ ktrue
-1666666666234567890
, C% {' A. J: P, |9 y0 e( D$ Qtrue
-1666666666234567890
5 J! }, R2 {. w2 g* ]" jtrue
[ -1666666666234567890 ]
[ -1/14 ]



3 `9 w9 l5 M8 \7 g* M; G: F, T1 Y


2 J* g5 t) Y/ D2 I8 Z
; V- X1 D" U" V2 B1 }- T" b4 S, W-1666666666234567890
# V& O) H$ L# U$ Y6 Y+ n% Q-1666666666234567890
9 C( M0 @& d) b- y: H+ Htrue
8 Z9 V: ?2 f0 k- M: N  Y2 n-1666666666234567890
6 [: Y- S1 n' q0 C+ Strue
-1666666666234567890
; m4 Y3 {+ a: i$ Rtrue
+ R4 |! X6 h' c3 e1 a2 z-1666666666234567890
true
2 Y2 H, W6 ?7 U/ M& Y* B[ -1666666666234567890 ]
! ^, b2 P2 A2 t# s  m7 k. U3 D. \9 s[ -1/14 ]

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
s:=0x12345678111;ss;
' P9 K8 x$ f$ E- E! Q! O4 g- Z
5 V* v8 ?4 B8 zsss:=Factorization(ss);sss;
+ e. Y- K( L3 {( o) }8 }( |sss1:=Factorisation(s);sss1;
" j! T8 L% A* ]9 f4 o" x4 zFactorizationToInteger(sss);
8 S% F/ u. k  I& V# PFactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
3 ^: K# s6 s; n6 ?SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
+ r6 Z4 E, s6 \2 C7 ?9 cSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制
/ d' u; Q0 |3 _* _5 i. y" }
12345678111
12345678111
+ o; T# {" u+ }# E+ e2 c[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
2 x; h  S! o: z+ V[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
3 h2 {: ]) E4 C% @" c$ d0 w12345678111
! a" [/ B% S7 N: M/ L1250999894289
1250999894289
) X# d, o: ]- y9 u% ?+ w# H[ 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,
# }5 P. O) }( {. K# T1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
2 O. }( I, ]! f12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
. q# K. n- O' e8 Y1250999894289

lilianjie

% \: `" N9 f! X
Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
6 C7 P: M. B: m4 J! `+ n  \7 }n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
( \$ M, U/ \0 {( U# O6 a5 B3 E
+ V' w: t( Z) C! `( h0 q5 b
8 e! V6 J& q7 J3 e4 o3 p4 @1 H) UK:=Z!n1+Z!n2;K;
  a2 k' w8 a/ y5 @$ q6 J3 t
( g- [, Q% Q3 \, M; vIsField(Z);      是域吗Characteristic(Z);环特征
5 G1 m! Z4 n0 e; H3 {IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
5 ]) \- o/ B4 ~% x& GIsOrdered(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。。。。
IsPID(Z) ;主理想整环吗
2 \2 C5 b3 @* Y& U9 M, i
IsUFD(Z) ;唯一分解吗
: _" q6 {4 L8 T. j* [- p/ d1 r* z5 bIsDivisionRing(Z) ;除环吗
  S2 N7 H0 R6 {8 ZIsEuclideanRing(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
MultiplicativeGroup(Z);乘群
" K+ P: f4 K7 n. tCategory(Z) ;范畴Parent(Z) ;父环
1 K& {0 ~4 d: v" jPrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
: l* J# B7 ?$ x# O" i3 |# d9 dAdditiveGroup(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

ZZ:=IntegerRing() ;ZZ;
ClassGroup(ZZ) ;
7 \' c6 g5 c7 |! Z" S/ i0 i
===========

& a2 x. M" R/ |0 ^8 [* WResidue class ring of integers modulo 5
# J+ G: G) N0 }5 H# T1666666666234567890
1
1
" D% b. w+ H4 O7 R2 p( a6 R; @2
6 P- M( `5 b. O; c! jtrue
/ m/ }2 k' @) @  m5
true 5
& O+ |" W( Y1 r8 P; V% O6 Htrue
false
true
true
0 h9 m0 I- j# Vtrue
true
" d. l5 ?: Y7 |& t& s5 Ctrue
true
true
Residue class ring of integers modulo 5
4 r% ]5 @$ F  O6 |' f7 \Abelian Group isomorphic to Z/4
4 p; K1 L+ X) C# [! M) BDefined on 1 generator
! M9 h; A  ^- v8 n8 A' {Relations:
    4*$.1 = 0
Abelian Group isomorphic to Z/4
, o6 B0 L+ q6 t! W; kDefined on 1 generator
  A/ w+ k* r4 }/ |Relations:
* v& r$ S$ P5 y% E+ s. v    4*$.1 = 0
0 i2 N' K! x9 e2 n( bRngIntRes
- x# ?" K* x1 i7 I) A0 n* e! FPower Structure of RngIntRes
& W8 P/ k# P9 i* V* D: @' VResidue class ring of integers modulo 5
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
Defined on 1 generator
) \6 K7 I* x+ U' ]" hRelations:
    5*$.1 = 0

' w4 k9 m$ R" T  e; p- Y; m; w>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
% G& P" j9 m( J4 x# }Argument types given: RngIntRes
( ?' D9 e7 b9 ]& ^1 O8 T5 j
6 _! Z) Z" v7 r: kInteger Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
% o2 ~* b8 u" N( p' H, x- Q+ \ UnitGroup(Z);
2 h3 @( @# j1 t% I; r2 p' ]MultiplicativeGroup(Z);
Category(Z) ;
+ }1 |8 r% t5 T, C( {& }# u# K  wPrimeRing(Z);
2 g0 R/ z' o4 n7 p6 F* v5 YAdditiveGroup(Z) ;
6 O6 }) A5 w! C9 M
! n/ `7 Y5 m/ j4 ZZ:=IntegerRing(13) ;Z;   
6 m# \  j* o1 ?* H$ K: [ UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
& R% T% X2 ~: y0 w3 H) ~- WAdditiveGroup(Z) ;



6 U( v/ l8 r( `$ O5 B. E2 h' AResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
1 b& X, c$ ~! Q; L" ]Defined on 2 generators
& S, z# `7 O4 Q4 x8 s! q# zRelations:
8 N# F- g& a, H8 R$ H    2*$.1 = 0
' j8 [# R2 @" N/ u    2*$.2 = 0
  Y4 n# M7 t' A( J! DAbelian 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
& l1 F6 }: H2 B5 [/ K6 rRelations:    2*$.1 = 0
. L" F  w4 a/ ?( O; D6 u: E" [- u; O    2*$.2 = 0
6 v1 X1 f  U5 W2 ]- S; e6 mRngIntRes
) Z( ^% x$ |" U) X7 Y1 W" E& b$ wResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
9 [7 d5 l3 Q: J" DDefined on 1 generator
Relations:
    12*$.1 = 0
- X2 S" M; d) B- ~9 U8 T: xResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
5 N/ F7 |' a; T% P9 ~2 ^  {  H1 IDefined on 1 generator
( w& h5 Q* e9 @, N3 ^6 @* Y: A; MRelations:
    12*$.1 = 0
2 ^( n% m0 b' a9 `+ x3 {Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
# v5 f0 c2 _9 W* I, z1 q; }% W    12*$.1 = 0
RngIntRes
Residue class ring of integers modulo 13
9 A2 G; e/ F" x# }  v+ t& s/ EAbelian Group isomorphic to Z/13
; J% {! \( j0 F6 jDefined on 1 generator
Relations:
2 ?$ i& [! Q" t    13*$.1 = 0

lilianjie

4 |+ R* X% b: u8 s, VZ:=IntegerRing() ;Z;   
1 D; t( _4 N5 E
8 ]; ~0 T7 _4 X0 fR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   
6 u' D* P. H: L& @( r

3 x1 C" I  M/ }1 {5 ~PrimeRing(R) ;
Centre(R) ;
3 y; s* ~; p1 z+ Z1 \
& K9 N' t' Z" W* Y5 w$ O. J" v0 D, HCharacteristic(R) ;
# R ;阶----元素数
- Y" c; s" T. M8 {: x2 iIsPID(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) ;
- t5 {! [9 M  r/ F  }1 S
- f+ M5 H5 `0 W& ]6 j1 p, bIsPID(S) ;
IsDomain(S) ;
1 U- S( U4 `' k: G) t% R% ?% IHas**(S) ;
R eq S ;
! }- S* N3 K! v/ G+ FR ne S ;
. t6 N6 p" \/ T  k4 g; F
6 P8 v9 }+ z$ D+ uParent(R!123) arent(S!123) ;
Category(R!234) ;Category(S!234) ;

7 F/ \4 }5 Y6 q; ]# Q6 S1 {a:=Random(R) ;a;b:=Random(S) ;b;
2 d/ D8 F# D) s; z7 W8 z* ^  ERepresentative(R) ;
Representative(S) ;
8 S3 Y& o" I6 k5 H( _, p
(R!a) in R ;
(S!b) notin S ;
" N& K1 o0 {; y7 `IsUnit(a) ;                是单位吗
, g2 V" d, w2 a( V) @: P4 ]IsIdempotent(a) ;是幂等元吗
$ M4 V1 g5 v$ J- O- X( b, }: dIsNilpotent(b) ;是幂零元吗
0 B) w) @% `" g' F, z7 u8 p% B0 lIsZeroDivisor(a) ;可除零吗
$ X' y( B2 I4 [9 }4 w+ p8 i" j7 EIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

9 B4 m( Q5 d6 n0 y! h2 DZ!a gt Z!b ;
Z!a ge Z!b ;
$ k7 ]- w$ Y" r/ mZ!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) ;
( N" Y4 }& ?- H: QMinimum(Z!a, Z!b) ;
Minimum(R) ;



Integer Ring
$ R. m0 [  @5 l3 `8 H& RResidue class ring of integers modulo 12
1 V4 R$ O  W& n3 A+ D* X! wResidue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
) s2 s0 v# W- V. |, ^$ d12
12
false
false
7 E2 a* F7 n/ Z! `. ~5 V3 |true
9 x2 O" @/ p: E! w$ A' i' v3 {* mtrue
true
8 r; S' I. x  \; ~# z9 O! _true
, s, J/ t) t# s1 J7 Z5 \false
true
* Y7 e/ R# R6 V3 s9 P# hResidue class ring of integers modulo 12
1 O) s8 s# a6 |' RResidue class ring of integers modulo 13
RngIntResElt
  k  q2 [. h: o: q5 Q" ?2 w, PRngIntResElt
9
! _/ X( k; c; L* r3 p12
8 K3 E) L- ]. L7 f- d" G  |0
$ L$ h7 {: o9 _' d8 c% S1 p+ l0
true
false
# t. x; B7 C5 r% h$ afalse
+ G' M1 p3 d) ztrue
false
true
3 `2 q$ m* r% u2 U* Cfalse
- m/ t1 w/ e. U* x3 Zfalse
false
false
, K2 B/ Y& K- k8 Z7 n6 Gtrue
3 |' b4 x9 Q' @: l  j; S. l- q7 jtrue
* q2 _8 r/ V) |5 r/ F) J7 [12
; g5 D( g3 F; `1

>> Maximum(S) ;
          ^
9 J/ F5 [) h6 I8 L! O$ g) d% L& \Runtime error in 'Maximum': Bad argument types
9 S: k# v& A: G2 N. FArgument types given: RngIntRes
* \+ I* Z, e) z1 ^5 Y# J
1 n8 ?8 J6 V2 Z9 U! k3 Z% _) `/ J9
) w& V  Z; e2 E) s5 ^/ B+ l+ u
& \0 F0 U% _% z, |6 O/ W2 R2 F>> Minimum(R) ;
. t4 N7 S6 J+ \4 W# j          ^
Runtime error in 'Minimum': Bad argument types
. b) w, D/ M$ b! n; p& F* M5 \Argument types given: RngIntRes

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
- g# p' l3 _7 r" _9 PZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
3 I8 O8 S1 i, F0 w( J& |IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
( o( U! _( [* L2 ?ZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
IZZZ5;
! m9 [- G( }3 n3 Y3 _3 z
I12 *  IZZ15;            理想和/积/并/交，
9 w' Y, H* ]2 d5 Q$ p" I) F理想和是理想对应两（可多个）元素加，
% e1 x  q& t4 m) y# {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，
" G2 }9 w: J' g! M
# _* y/ {& s- e6 Q理想积是理想交的真子集，极大理想交是理想------J根
/ T4 ^0 N) s) e0 S9 O( `理想商就理想间同态：是必须能整除
/ {! C* A% w* z" s7 Q( nI12 +  IZZ15;
2 |. V, M' J# ^/ w, @I12 meet  IZZ15;
0 Z- w7 M( O( C. A* F
I12 * IZZZ5;
I12 + IZZZ5;
) d. @9 U3 ^: J* PI12 meet IZZZ5;
I12 / IZZZ5;
' b# I' z8 l) V$ e# `& rIZZZ5/ I12 ;
Z * IZZZ5;
I12 + IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
* @; c# @+ r. o( s# qIZZ15 meet IZZZ5;
/ m/ `+ B1 x8 x; F( hIZZ15 / IZZZ5;

& E8 U: s, `; M' {, X' `' a* BI12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
+ j8 ?9 Q" W/ V( A0 y8 mIZZ15 subset IZZZ5;
$ }4 H# Y1 }1 D+ ?9 O( e4 yIZZZ5 subset IZZ15;
Integer Ring
/ A% a: G' A3 _9 |- w9 @Ideal of Integer Ring generated by 12
  r- a# d/ L; [0 o0 mResidue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
$ g0 b$ X$ [: E# F* u2 `. }false
Residue class ring of integers modulo 12
, h/ Q& P7 e1 s) [! w+ qResidue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
/ [) f5 F3 {& m0 eIdeal of Integer Ring generated by 180
" F) w1 M: o$ m+ }* T4 _8 LIdeal of Integer Ring generated by 3
6 o7 P" u5 B# }4 [3 j. R3 |" hIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
Integer Ring
Ideal of Integer Ring generated by 60
: K. F% s# \$ V8 T  E3 B
>> I12 / IZZZ5;
; e; Q& U* R: w2 u$ u       ^
# C3 {% J; p' P/ I' ]Runtime error in '/': Argument 2 must divide argument 1.

9 n" p/ k* m' C0 \
( k  _% Y" |1 S( T! W>> IZZZ5/ I12 ;
        ^
Runtime error in '/': Argument 2 must divide argument 1.
( U+ @1 K# O9 f9 v* }0 g2 ~% C! w
" B& V' y, D$ O" O# f* t. c% _Ideal of Integer Ring generated by 5
Integer Ring
8 a* d6 I# F1 h- ^Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
$ f9 ~, n& a4 C- x: s6 z7 aMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
$ @$ L$ p5 ?5 S$ \9 JIdeal of Integer Ring generated by 5
8 z' X: o5 W# w( f/ f% t9 sIdeal of Integer Ring generated by 60
# j& |; Y0 N/ q& @6 `# R0 C* M! rIdeal of Integer Ring generated by 15
" t& r1 o" W; h8 N' o# bIdeal of Integer Ring generated by 3
: C  Z) S, p- }. J) C) q" lMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z

# u3 @: {* g) h, rfalse
true
, K+ ?3 i1 f/ S4 P8 jtrue
false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
I12;
5 ?) p: w; d, x  O. I- ^/ e- PZZ:=IntegerRing(60) ;ZZ;   
4 W" m, N+ a, l3 b- bIZZ15:=ideal< ZZ | 31 >;
IZZ15;
1 k+ s; \! o8 f0 q5 G! ]ResidueClassField(I12);
7 |' g3 D) l  c4 I; GResidueClassField(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
2 [9 M5 h& J! z# v$ sloc< Z | 19> ;
loc< Z | 17> ;
loc< Z | 131> ;局部化：一个素理想到原环元素的映射
8 C* N' o1 Q! W# Y; j1 |1 Mext< Z | > ;超越扩张到一元多项式
" V6 [$ l$ `$ I. o9 Y! b6 Mext< ZZ | > ;
* n- |3 C/ m! [) }6 g$ U# a4 ?
ext< Z, 2 | > ;超越扩张到多元多项式

: K6 \; X! D/ f4 G1 C# Z! P, d% [5 Dext< Z, 3 | > Completion(Z, I12) ;
& \: M2 j7 N+ c3 N: z% u

4 y" S0 |( d4 i( Tcomp<Z |I12  >;
    素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
comp<Z |0  >;

. a' X$ {: C' eInteger Ring
Ideal of Integer Ring generated by 13
2 G+ h; U8 i( ?: R! }9 JResidue class ring of integers modulo 60
3 @# B. S! H' X, A% j0 VResidue class ring of integers modulo 60
: C, r' J. w+ r8 w2 a4 W' V# C1 fFinite field of size 13
Mapping from: RngInt: Z to GF(13)
# E& T7 L6 w9 p, i/ _modulo 13
9 T4 ]( h! @$ V
2 P. T- r0 u8 d  x9 ?  r+ n# u>> ResidueClassField(IZZ15);
7 _# n* V3 t9 R/ [% G' \                    ^
4 Q" V6 v+ T/ y+ L+ SRuntime error in 'ResidueClassField': Bad argument types
3 \- f" L  V: {Argument types given: RngIntRes
% d4 ?" r, L4 R; i
Valuation ring of Rational Field with generator 19
7 _& T1 S6 n9 ?. T6 o+ C. DMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
Valuation ring of Rational Field with generator 17
6 n( a* ~( t) ^6 W: [5 t* S& NMapping 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
6 {5 [9 D) V1 m9 Z* V! R! L4 |Univariate Polynomial Ring over Integer Ring
3 Q, ~- I9 o9 e5 i$ U) ?. yUnivariate Polynomial Ring over IntegerRing(60)

>> ext< Z, 2 | > ;
      ^
$ o3 N# `* G# {4 s4 k' rRuntime error: This constructer is no longer supported
( m" ^, S3 k  D/ w' b: G

8 [0 @+ @( z- P+ J! [>> ext< Z, 3 | >
" U. g' b" n1 X  ?6 ?' P      ^
Runtime error: This constructer is no longer supported

' T& j- A  J, S/ f+ S' @" Z& _2 x13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
! `. C6 N# p2 [, }5 }
+ @, Y4 s9 W# c- fCompletion(
9 D- ?# H; ^; j$ C9 H    Z: Integer Ring,
6 C: I; o' I1 i. B- l2 s    P: Ideal of Integer Ring generated by 0