一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
> n;
# R% y2 @, c$ U- t( L
> n:Hex;                           转16
5 d7 w0 P+ S* c5 zIntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
# H5 s( U: U3 N; w' o8 S3 f# T) Y' YIntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
Representative(Z);         环代表元
Eltseq(n);                        取整
( q2 [: z* K7 v4 I) R/ J7 {Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
7 @1 q4 }/ u, |
2 s. y8 @( A  v$ H. ^' r0 Tm := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
5 i9 V+ g" O7 I2 p2 H& y# t6 Bk := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
- M$ M$ ~, g/ w- Y9 ?> k;
n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
2 M8 z' q( n, a* Q' h- [/ ?, x> kk;
- t" P) E4 `6 s1 V% {kk eq k;
' Y5 U4 M( Q7 H. Z7 S
+ z/ W9 B3 q3 I3 V1 ]k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
1 `0 R: Q; m' ]* i6 V> kk;
kk eq k;

2 A4 {  N$ O! C  U! T+ IEltseq(kk) ;Eltseq(-1/14);
' ?7 r, g0 r# n3 U4 c0 t
- A* P3 W  [7 y) s

" i5 f6 n+ y. ^) o4 D
. {8 R( C" I% X/ M) O1 j# Lk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
6 K. z+ f; Q0 m# G" U1 m# z# \% ln eq k;
9 K: }8 X0 f. o0 G, s& N8 |* {& vkk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
/ v& y4 v/ |. y> kk;
0 `. S; o2 p/ S/ O4 u7 g0 ekk eq k;
+ {! ?6 r% C& P; w: a2 {3 h
2 ?8 R2 i) R- e9 s8 g  Ok := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
9 q0 c  w8 o5 i' f$ e) _n eq k;
+ d6 B$ E% g+ m% J! I. F0 akk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
6 I1 p5 J- X- d2 u> kk;
9 y5 i# k( \( ?  u- s: c  F3 I' Vkk eq k;

Eltseq(kk) ;Eltseq(-1/14);


# `) l5 A6 r) d  ?4 [
) r0 t' ~; _6 X! j( A
: }/ g& h8 Z7 O: N# [, v" k! g" X8 Y1 ]
# k' a" q$ j2 n4 G
0 d$ \" c. M2 c( T4 M0 X  j5 {


* t8 v3 A; S. ^$ w0 {: n3 b
* o+ B' J- n4 b' s: U; D=============
) N, O& m7 @4 |7 @


+ ^$ a( [4 B# v& y7 L
* @* l# Q1 }) l( H( jInteger Ring
9 ], S" Q0 n; B  ^: X4 Y% R% K-1666666666234567890
5 \& X4 C6 l. @& G9 p; q( j-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
) \$ w' F# i/ ~! A" E0 y* S! c-1666666666234567890
0
! l! g6 {" Z* Z1
0
[ -1666666666234567890 ]
2 O$ ?7 Y" s8 S/ Q! I. s[ -1666666666234567890 ]
1
13
1
# n5 c4 O. ?! G) j/ _* F
; u) F1 {# G  H9 f% J-1666666666234567890
7 |1 z7 y: H7 p7 y& h9 ?true
. f/ ^! P& P! M; R- o1 @-1666666666234567890
true
-1666666666234567890
1 s9 @( @* Q1 G/ _% W$ b) U; Otrue
3 U" b3 A! R: x& i-1666666666234567890
true
! D3 o( T, S1 V* T) }- d[ -1666666666234567890 ]
+ M4 M: u2 `3 m& E& b( k2 `[ -1/14 ]

, _2 C1 x/ ~; b


3 K- l0 d5 c6 b8 L5 \8 q
/ d; d1 ~- C, I0 I3 P; _# S  p

6 r: y/ d% f  L8 W-1666666666234567890
-1666666666234567890
true
-1666666666234567890
& ~  G; R# Z4 c; F" N& _/ @true
+ T  @6 y8 [1 p) `2 B-1666666666234567890
8 a. D( T& O$ w9 Wtrue
-1666666666234567890
1 H3 c) `' u) q0 U4 atrue
[ -1666666666234567890 ]
[ -1/14 ]
. m+ q  P5 Z0 a) c

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
6 [' K9 Y. }7 ]. @5 R! Ts:=0x12345678111;ss;
' x8 Q$ c( X% m
5 {5 C" q& u/ j& O# \! C# osss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
+ E9 {/ Q9 w: p7 oFactorizationToInteger(sss);
; o& x: b6 i" e! `( c) i' R+ j6 p' mFactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
0 p1 x0 x8 y! p8 g+ G( C, Q' Rssss1:=Intseq(s, 17);ssss1;
" g' \# ]+ D( `) s1 w# sSequenceToInteger(ssss1, 17);转成2和17进制

12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
$ x8 g; J" i0 A6 ^[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
0 U8 i$ O$ J6 s1250999894289
2 S! u/ F* G1 A. o5 D4 i% o2 r0 J[ 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,
1, 1, 1, 0, 1, 1, 0, 1 ]
" g( I3 q6 J5 o. O) l12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
' j% F& T- N$ W1 v12345678111
! S" R2 e, o  l7 H' f+ }# D[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

lilianjie

/ N2 N9 E5 y8 i( P/ |
Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
8 q% t: J* E: ]  ?/ N1 U, Z. P9 Y) W> n;
/ P4 x- W( {8 m5 t4 C7 v! hn1:=Z!1111111111111111111111;n1;
! A+ S( a5 b9 Q" e5 h7 f- M8 ]n2:=Z!11333331111111111111111;n2;

+ k" p. c- i5 ]
K:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
9 i) U4 C: g. `+ t- ?' PIsOrdered(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
* |  \, Y+ l. [% e2 h
  e- ~0 B) y) j+ r! A2 ^IsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
4 k* E4 Q7 v# H# w) o2 s8 y7 ^: aIsPrincipalIdealRing(Z) ;主理想整环吗
& n9 ^0 l3 E4 l: r! }* fIsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
# A7 q9 b  |1 d7 V$ iMultiplicativeGroup(Z);乘群
Category(Z) ;范畴Parent(Z) ;父环
% n. R8 D6 Q: ?PrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
AdditiveGroup(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
  ?) V+ I8 p% q
" y0 P6 V9 M7 G# T3 tZZ:=IntegerRing() ;ZZ;
+ D% Q+ v# M3 \# N8 jClassGroup(ZZ) ;

( ^. q0 Z- V6 K' y( `6 v/ \===========

Residue class ring of integers modulo 5
1666666666234567890
4 d5 H0 I; _2 g1
1
( c0 ?% X% Z: \2 J2
true
5
true 5
' F" D6 R2 l; P! f; y6 ?+ Ktrue
false
true
; f' {1 g, V9 X& ^4 u$ u- U' o7 gtrue
6 H6 y9 _8 T) m8 N& Ftrue
true
0 i1 Y# @- c' U7 Z$ O. ztrue
true
true
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
9 H4 s8 s& R: }Defined on 1 generator
6 k& M" B9 ]7 d4 Q* q4 ?7 @# _Relations:
    4*$.1 = 0
Abelian Group isomorphic to Z/4
) F7 [0 w- Z! ?' YDefined on 1 generator
Relations:
1 x1 m5 @1 I% }! }& Q$ n, |' u8 D    4*$.1 = 0
RngIntRes
- _3 U' h" l+ Z4 wPower Structure of RngIntRes
Residue class ring of integers modulo 5
/ H/ f& F' o: xResidue class ring of integers modulo 5
6 M  C' _7 J5 o( R! B* xAbelian Group isomorphic to Z/5
, [) e# G# T% i- o! R0 F0 sDefined on 1 generator
Relations:
* l. L8 f) t( ?9 S! B    5*$.1 = 0

* E8 X: @* v1 u. ?" k, k- B>> ClassGroup(Z) ;
- u- W5 G9 m1 Z5 _& ]6 p             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes
% z9 W( b4 ~' l; m9 X' g
6 L0 g- j6 W: w# vInteger Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
# U# R! F* |1 i, j. Z( |7 AMultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;

7 c' m3 ~; q9 HZ:=IntegerRing(13) ;Z;   
5 Z5 ^3 H4 t$ k) t8 j  j UnitGroup(Z);
" |+ v8 V( L* z) x2 H* IMultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
9 O3 g8 [  q: Y' T: ]; t! r( S% J9 XAdditiveGroup(Z) ;
( M3 H8 u! \0 A, w$ N' m6 B
- _" w0 M* q1 _, W- P. e
" a/ A8 H4 P5 \$ W6 D, Z
Residue class ring of integers modulo 12
, H! f( {4 D0 TAbelian Group isomorphic to Z/2 + Z/2
4 d5 o* e3 G+ W6 z& UDefined on 2 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0
Abelian 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
* L) l, S8 {3 J. }7 WRelations:    2*$.1 = 0
$ F$ L7 \/ J  m: E9 \    2*$.2 = 0
6 S) z" |7 E( {2 c8 l; e$ _RngIntRes
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
" g% L8 w1 U" J. z$ pRelations:
    12*$.1 = 0
2 C( _" C8 S' U0 `- OResidue class ring of integers modulo 13
& z$ g) b/ g6 U3 v7 ]Abelian Group isomorphic to Z/12
Defined on 1 generator
& R" n$ g% i& A( r) C2 nRelations:
    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
' }& `1 Y1 u8 Q2 q( `8 P7 XRelations:
    12*$.1 = 0
3 m; j: p) I! N  b2 Q) qRngIntRes
' m6 c0 r4 b3 V4 h) rResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
Defined on 1 generator
Relations:
    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   

R:=IntegerRing(12) ;R;   
, x' x* E$ o+ k" k0 W5 {0 \' U& D2 tS:=IntegerRing(13) ;S;   
4 U) F. ^$ A' ~: F: n' B% V  g

PrimeRing(R) ;
, ?$ ^2 O3 b* e; d2 i5 I+ RCentre(R) ;
5 S: C- Y* S5 ^+ V2 ~
$ _- E: G: b+ ?5 i( yCharacteristic(R) ;
# R ;阶----元素数
. ?$ w/ g3 d. l' q$ j6 |7 xIsPID(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) ;

5 c; R5 _( W4 X7 U  \2 i# `IsPID(S) ;
1 G; G5 e, b; [1 ]# z$ |3 iIsDomain(S) ;
4 T* C  R3 h* Y$ y2 `6 PHas**(S) ;
R eq S ;
7 R$ i6 ~+ b0 S6 b$ Q& e& q3 R, eR ne S ;

$ V' t( E; {2 v% b. J* J2 O8 ?Parent(R!123) arent(S!123) ;
% V" b; f7 I1 l' K/ o. f# jCategory(R!234) ;Category(S!234) ;

. w: k) L3 Q- _a:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
# o2 _. e' M  z3 j1 o2 GRepresentative(S) ;

# e" S" ?6 k5 Y8 L' h1 l(R!a) in R ;
( D1 B/ t+ ]9 v2 X2 f+ G(S!b) notin S ;
# E. J0 D; {* Z  t8 n) {IsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
% O  K( q9 [% hIsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
8 _( I: w; k. a$ [* }& V" {IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

Z!a gt Z!b ;
  t! ?8 [, r" F* ~+ L, wZ!a ge Z!b ;
' W: f9 F& |1 [Z!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) ;
- y% k# Y4 \0 z! ]4 C& ~Minimum(Z) ;

, ]! M. y6 M' q. B1 u7 |Maximum(S) ;
5 V1 y  ~/ f6 o; t% K, a3 U) T! ~" t! _Minimum(Z!a, Z!b) ;
6 s# U3 c" P, S6 v' u0 z. B9 s% HMinimum(R) ;
8 Y, N6 k. \4 \$ W" M* }. ~1 `


Integer Ring
Residue class ring of integers modulo 12
4 q7 A6 C) N. `% m& ~2 VResidue class ring of integers modulo 13
: ^6 K; ]* c. B/ ?& ~Residue class ring of integers modulo 12
5 C4 n! D. \  Y6 }Residue class ring of integers modulo 12
+ M8 J1 r; u- L; Y, h& v/ i12
12
0 ~* q; q; \2 D0 Z9 R# [1 rfalse
( \$ f" q# S, nfalse
& e$ F, E$ v" Jtrue
true
true
true
& c# i7 I2 J( O1 f6 |6 B6 d5 w, Ufalse
8 T4 F. o& q! p$ K# \% {  gtrue
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
9 c1 x# l! x( `0 \RngIntResElt
RngIntResElt
2 U' [$ }6 a: M: h) I# z' w9
2 ?# l' X- z9 q9 ^12
0
! d# F2 B: k: ]4 ^0
true
false
7 n7 x; n8 W) s8 a! _false
true
false
true
false
( X7 \# W4 K; x* r! Q( V0 l9 a, l. tfalse
( d" q& V2 ^6 F3 P. a2 _$ a# [false
/ \1 X+ x7 F: q! W, c0 [9 k3 Jfalse
# d  i7 Y0 c0 u, wtrue
true
12
# T$ v+ }' u  g# ~: i! q% n1

- O3 h  h4 b2 O; X>> Maximum(S) ;
          ^
1 n6 d" V5 N$ h3 z5 e( n& F: u. L) I% ARuntime error in 'Maximum': Bad argument types
1 M1 C, T0 k# B% p4 t4 QArgument types given: RngIntRes

# T6 W" \; F' j4 A8 e% f- D9

5 o: d% a: F1 M5 r, s7 x8 z>> Minimum(R) ;
1 r3 b% m+ a: y. g8 P4 Z( ]          ^
Runtime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

lilianjie1

; n3 V8 D8 ]1 Q; u; e% i' ?
Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
ZZ:=IntegerRing(15) ;ZZ;   
3 `, M5 D( A3 H+ I2 g) ^  yIZZ15:=ideal< Z | 15 >;
( B; ^* O7 H1 v* U3 g2 sIZZ15;
- x! f, z4 K" L' {I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
( W- l# P& E7 ?# ?ZZZ:=IntegerRing(5) ;ZZ;   
  D6 @. J8 s8 OIZZZ5:=ideal< Z | 5 >;
IZZZ5;

I12 *  IZZ15;            理想和/积/并/交，
/ E5 q; g8 g- Z理想和是理想对应两（可多个）元素加，
理想积是两理想（可多个）对应元素积，
% U1 G8 [& f' s/ m理想并就两（可多个）理想元素并，就不一定还是理想，
" J+ I8 ^/ _1 D" l' ?/ z+ i3 ?1 A$ g& [理想交是理想（可多个）元素交，理想交一定还是理想，

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
2 d" Y4 ]- m7 r6 y/ c6 v5 _I12 +  IZZ15;
. A- [( u# b+ TI12 meet  IZZ15;

! h- N, h; k% T; aI12 * IZZZ5;
2 M+ b; B- W$ HI12 + IZZZ5;
I12 meet IZZZ5;
7 ^0 T" `0 i  p# V- V/ E* d  rI12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
9 j! |% y/ B) SI12 + IZZZ5;
* A, Z2 A+ {+ oIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
IZZ15 meet IZZZ5;
7 b$ T8 E, C( e! \7 I( n$ c- {IZZ15 / IZZZ5;

I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
" h" l5 n4 l. t# ^4 aIZZ15 subset IZZZ5;
IZZZ5 subset IZZ15;
Integer Ring
0 n, ]+ ~; h% u3 S( G; f3 OIdeal of Integer Ring generated by 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
' H  R% l7 u' [! _7 Lfalse
/ Y, q" |% s  t. F0 r+ I: Z- ZResidue class ring of integers modulo 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
: [1 v- M( z0 z# J" J6 e7 h" GIdeal of Integer Ring generated by 180
& g0 V) Z5 l) J* a  Z' ]9 `6 d; o4 WIdeal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
+ |' n: d* y* w% I- uIdeal of Integer Ring generated by 60
Integer Ring
" Q) c3 \: E. W) N( y; Z4 R) m; K5 \Ideal of Integer Ring generated by 60

3 n; v  g& z7 T' U; y: R' \>> I12 / IZZZ5;
3 Z& t. Y6 u& a       ^
  E" E0 y% G7 Y6 A9 s7 R1 Z+ cRuntime error in '/': Argument 2 must divide argument 1.
3 d3 S/ G  i6 U

7 I9 D$ K2 ~8 `9 E+ k  |# A>> IZZZ5/ I12 ;
        ^
& V" B' m" D, b3 wRuntime error in '/': Argument 2 must divide argument 1.
+ \" b/ A: B3 C- [4 D
( ]' P9 Q# o: tIdeal of Integer Ring generated by 5
Integer Ring
9 i! e7 }; X4 E" Z- ]3 w+ ^8 X: MIdeal of Integer Ring generated by 15
- _- g% w8 h: |) x  q5 j" f+ LIdeal of Integer Ring generated by 3
Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
Ideal of Integer Ring generated by 5
Ideal of Integer Ring generated by 60
+ U+ L2 b1 ~. ZIdeal of Integer Ring generated by 15
1 Z1 F5 j% p3 W% v0 _: S. GIdeal of Integer Ring generated by 3
+ l% [# V! m& `9 jMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
  |! S7 s! N1 W, v
false
" ]/ t( h+ a& G( x2 }" ytrue
true
false

lilianjie1

Z:=IntegerRing() ;Z;   
# z* E: W% V# z/ z: ^I12:=ideal< Z | 13 >;
1 x1 \( U- y# E" zI12;
8 r3 u$ E" a# M( j) e# }1 E  S4 k' zZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
! L0 P6 N6 e4 B' iIZZ15;
ResidueClassField(I12);
  g' z3 E* b+ o$ T" hResidueClassField(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
5 J6 r! u3 ^) d: [& U8 \6 v8 c6 Tloc< Z | 19> ;
# R% y+ @$ y' `; wloc< Z | 17> ;
+ n9 C8 I# a. o# bloc< 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 | > ;

/ B* ~, N( x2 H9 E4 n. Wext< Z, 2 | > ;超越扩张到多元多项式

ext< Z, 3 | > Completion(Z, I12) ;

6 _$ m6 `7 q& H& G* W% W# H8 h
comp<Z |I12  >;
    素理想零理想完备化，和P进环联系起来
6 M/ Z% G0 ]& t6 R8 dCompletion(Z, 0) ;
comp<Z |0  >;

Integer Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
Finite field of size 13
- N) i  K2 d: K3 I/ xMapping from: RngInt: Z to GF(13)
modulo 13
2 O+ Z: r9 d# H( f
>> ResidueClassField(IZZ15);
                    ^
Runtime error in 'ResidueClassField': Bad argument types
$ @. k0 I' A$ g2 `8 j: h; PArgument types given: RngIntRes

, m: X& z; f6 l4 p3 GValuation ring of Rational Field with generator 19
% k4 c" O/ e  ~& V. m, {. u1 OMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
Valuation ring of Rational Field with generator 17
+ P$ j; h/ A7 d2 z9 R/ W# K  {Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
! b9 x; K* U; ]9 g. U1 Y! W/ XMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
/ C% t% i4 p1 `" YUnivariate Polynomial Ring over IntegerRing(60)

4 G* N1 |# y! S$ \# x>> ext< Z, 2 | > ;
9 ^; N/ y4 ~4 K! v$ w      ^
9 g3 O! `4 s( @% D; bRuntime error: This constructer is no longer supported


>> ext< Z, 3 | >
      ^
0 c( Q- F1 p7 w) Q- fRuntime error: This constructer is no longer supported

13-adic ring
1 u) b( v  ?: H+ S: ZMapping from: RngInt: Z to pAdicRing(13)

; A. k" ~- J, j; |, i& _( bCompletion(
7 @7 w" v4 B4 T0 D4 e" |) I    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0
& `3 C0 K2 e' c. h2 N. X