一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
) b) _9 v. w' \+ X2 R> n;
3 Y5 _+ E/ [  o1 P' p- n
> n:Hex;                           转16
IntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
$ ~- u/ l' p$ rIdentity(Z);                  
2 J+ g5 [; ^' \Representative(Z);         环代表元
; R( @- {8 \# E) {( q6 j! zEltseq(n);                        取整
/ O5 V# ~6 B. n) yEltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);

7 S; k8 B. }% z$ U# Um := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
5 \4 F9 w7 J( r3 j0 X5 Xk := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
" t) `" t' i9 V+ R# G> k;
: V. E2 m4 l+ bn eq k;
8 V2 d3 G: B0 i, skk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
1 z$ o/ U+ c: o* _+ b! A0 ?: e
) L4 b3 k+ z# V6 `. L- Gk := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
2 |6 U& r" ~) @' \: G: S' x> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
5 ]7 i9 w/ b0 U; J' w> kk;
kk eq k;

2 W6 T. ^5 W: `1 T6 J+ E; lEltseq(kk) ;Eltseq(-1/14);
- z& `+ @6 M- S; i; |# m- f' P5 T
# t6 @0 X( m1 l' \4 x
* B# }' [/ N) k

k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
4 ~/ U( d$ l/ V. `n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
5 L8 B/ |# ?+ y7 `( E> kk;
: Z$ h, H3 z# I/ l& C/ G4 `/ ?kk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
. j7 ~* g. j5 D- Z5 _n eq k;
2 p. A1 w  Y1 @) dkk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
1 }" v. L. J( R2 n* nkk eq k;

- h  Z) e7 @# hEltseq(kk) ;Eltseq(-1/14);
! D$ T5 t' H" V  o7 b. ?' Y

1 u! A7 ]" i9 h% \
+ ^# L5 D" U; @  d. d4 A: R9 ~


2 E- J5 a( S7 o+ H& J0 r
6 S9 ]( D- p( L% z2 D8 |
, I8 v  X( a* z0 r) v" O/ M
$ O; n; \) C1 ?# f# T9 Y8 u
: h$ f7 q5 g! a$ K" V=============

1 ?$ v. }$ P" r4 y+ ]2 N


Integer Ring
/ m3 K" l: \# g9 ]7 S1 V& M3 t-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
* U  `" K. n- N, P, q  k2 [-1666666666234567890
-17213080A7E55CD2
( Y7 m7 r( `6 x$ ^: b% s; x-CNUO0WGPY9CI
2 \8 [7 j# {  ]( }/ S4 ^& \-1666666666234567890
" ]8 w( N9 {% x-1666666666234567890
0
* P% o/ B8 G, l1
1 e$ g+ F4 W7 }0
[ -1666666666234567890 ]
" m( A7 f& y5 Y* K. t2 b" G[ -1666666666234567890 ]
& T3 \9 |  D) ~0 t  a' r: s1
13
1

* j& i" p( }) c0 ^6 c6 c-1666666666234567890
$ ^4 J( W* B' W* t2 G! Ytrue
-1666666666234567890
true
; Z2 }# s, \$ V! s-1666666666234567890
true
-1666666666234567890
2 q, \' c/ f3 etrue
[ -1666666666234567890 ]
[ -1/14 ]
5 a2 n4 ?4 M( q9 `, Q- F( ^3 I
' o, Y. Y, l1 `' B+ `) ~% l

. @9 E. t4 Q0 [. s" K2 q



-1666666666234567890
) s% J9 P, ?# h7 U-1666666666234567890
true
4 W1 s/ E, \. t, V1 K* q-1666666666234567890
3 j* W/ H$ K7 O8 F! V$ o7 Ctrue
; r- @% i+ ^) @3 J1 e0 l( z4 B-1666666666234567890
' {8 I  [/ D  ^5 Utrue
-1666666666234567890
( ~& o: [5 o9 @& qtrue
  o$ e" C# Y& H& o[ -1666666666234567890 ]
[ -1/14 ]
% [  k  h1 I. ^
4 ~* g& v  X" R$ r2 H9 \0 D

孤寂冷逍遥

lilianjie

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

, Y- |' f& z; [1 B7 L) b" Z# Csss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
( H0 S& n+ r0 s- K) ], ISequenceToInteger(ssss, 2);
: n, ]" r% h! ~2 p$ t) Tssss:=Intseq(ss, 17);ssss;
6 ~  L4 a# [, J2 o# u+ M5 s$ r/ kSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制

: |" ^6 \7 d) e12345678111
12345678111
! J7 Q, T4 Z+ z2 r6 H& l1 y# t' i[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
3 B+ p6 t3 J$ \" A/ o: s- {[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
( y0 l. D6 w) P( Z: b/ p1250999894289
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,
1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
7 u9 m& O8 R2 @! h' G[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

lilianjie

7 Y( K6 y3 k4 b, v" d  a4 e0 U8 v
Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
5 c# N8 i) B9 j9 i> n;
* O4 X. P- ]7 n+ ~. c( ]n1:=Z!1111111111111111111111;n1;
& a8 b8 v2 P; z# A8 c* L' l, F2 Hn2:=Z!11333331111111111111111;n2;
# z8 G; m: v5 f/ h) `9 l. p

; {/ w  t$ s, v+ s3 yK:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
; O, C+ c. r+ ^5 XIsCommutative(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。。。。
6 J! d% P2 W% d; ~IsPID(Z) ;主理想整环吗

IsUFD(Z) ;唯一分解吗
/ z: Z2 r3 Z+ n7 S. U& H1 k0 G8 u$ eIsDivisionRing(Z) ;除环吗
0 }% ]6 A' z8 W8 L. gIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
; Q7 B7 W4 x4 X: n& O7 \IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
  y8 b1 `3 y3 @( ]UnitGroup(Z);单位群
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
' C. M% |+ r) |+ U+ I& F* s# C* IAdditiveGroup(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) ;

8 p; i% ~( a" [: V4 G  l# x8 o===========
2 J( J6 Z2 i; i6 ~, L9 r
Residue class ring of integers modulo 5
1666666666234567890
* {: V9 K! `( z. W5 M$ _; X. y6 \1
1 }- w4 }4 k+ _0 n: g* S/ n1
2
true
5
+ F( n: I  T3 X. X' F8 {  z* m8 wtrue 5
true
false
, R( \4 e. @  U* H9 [& ttrue
true
$ ^1 e1 Y3 G" y9 |2 C' ftrue
true
true
true
true
Residue class ring of integers modulo 5
& p8 V7 f1 V! {0 X/ W4 M/ yAbelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
+ f2 g. T$ R0 Z, h; B3 r  q6 T! ?    4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
3 O) O: s( c! \    4*$.1 = 0
RngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
1 X" {9 o4 G$ c8 K" x! _Residue class ring of integers modulo 5
1 K0 s+ n& n2 G. T  _; rAbelian Group isomorphic to Z/5
( C, M6 Z) E) r, V! G. K3 CDefined on 1 generator
5 W. v* z2 q' i8 E: W9 IRelations:
    5*$.1 = 0
" }; d, G& V) i5 z5 L. v% [
>> ClassGroup(Z) ;
7 X! T& q! V5 D5 z; Q& R# p( w1 l5 }             ^
% g3 L) p9 x2 j3 ?7 yRuntime error in 'ClassGroup': Bad argument types
2 K" z, W% Z1 p$ c  j. ^$ IArgument types given: RngIntRes
0 a& b! U6 e- `& w: f7 _2 J7 D
) ?+ f7 b! Q. b8 |Integer Ring
$ V8 W, Y/ j7 i& W, ?" s. _% |" R: ]8 IAbelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
: X9 \% R  a5 ?/ s% f5 MMultiplicativeGroup(Z);
Category(Z) ;
  a. O, E" Q8 F1 |4 G1 W  BPrimeRing(Z);
* u) W; f3 ~" b; aAdditiveGroup(Z) ;

Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
* b! p1 s7 v8 [3 _) W/ N; p  PMultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;

0 ]# |; D& c$ Q( W

6 ^3 g$ Z& Z% cResidue class ring of integers modulo 12
1 V; `2 u- F0 k0 Q- ~/ VAbelian Group isomorphic to Z/2 + Z/2
# _9 }4 r; y- Z8 D9 gDefined on 2 generators
Relations:
    2*$.1 = 0
4 l- I6 B) n/ q- u    2*$.2 = 0
  U4 }8 I/ `$ |$ v% H: u4 nAbelian 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
( M* M5 ~7 e7 b, o+ A& V7 XRelations:    2*$.1 = 0
/ t+ W' ^2 k9 }+ R2 k: L3 ^6 ?    2*$.2 = 0
* l/ l) G# c1 R- x7 a: r7 `RngIntRes
& Z0 o4 V; |; e, W4 \0 lResidue class ring of integers modulo 12
" j$ L7 O) S9 ]+ A* s  xAbelian Group isomorphic to Z/12
Defined on 1 generator
# ]1 {5 Z# Y& I" G1 aRelations:
- |, f- r; E6 M0 b% Z    12*$.1 = 0
Residue class ring of integers modulo 13
! K! ?# H  H* H' @4 g% MAbelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
    12*$.1 = 0
' Y) ^/ I  v/ TAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
( ^9 G6 ]: N/ B    12*$.1 = 0
5 O) r8 t4 n) z- D  l! T! [' `RngIntRes
' Z/ U3 P2 S4 x: F' `1 |Residue class ring of integers modulo 13
' V% P  h$ P9 X# g3 gAbelian Group isomorphic to Z/13
; W! F; s# O7 |Defined on 1 generator
2 z! o" k2 w  Z- C! p0 BRelations:
    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   
! g1 Q" \& X' ]& A! h0 M1 v
R:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   

* `$ I: }% ~! M6 ~# }( R. Y
PrimeRing(R) ;
Centre(R) ;
8 b. w9 i, H4 `" P( i, {. k
3 Q9 a2 F5 g, S: q1 F& jCharacteristic(R) ;
# 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) ;
% t5 z) D# Z7 X! x# ?& P  YHas**(R) ;
5 ]; l0 {6 `$ d3 m2 B  F
. o3 Z# n6 U. N/ O( }3 ZIsPID(S) ;
8 C+ V" W5 |4 v/ E4 S- r! VIsDomain(S) ;
1 x  g6 v4 j7 |" |  c4 ~Has**(S) ;
R eq S ;
R ne S ;
: E6 z! ]# G4 Y5 K3 p' W! J4 l
  @+ x; M$ k' l" f& _Parent(R!123) arent(S!123) ;
) f6 X/ {8 Z3 ]# ^8 s$ ACategory(R!234) ;Category(S!234) ;

a:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
Representative(S) ;

(R!a) in R ;
6 N: v1 `$ |$ ]6 c7 v(S!b) notin S ;
+ g! Q) ~; w+ w/ Y7 T# PIsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
9 n' a$ f" Q* JIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
( i: l' U% l1 l2 Z- b
Z!a gt Z!b ;
Z!a ge Z!b ;
Z!a lt Z!b ;
! F, D. W0 V$ HZ!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
! r  s% B& E/ N& i+ j$ MMinimum(Z) ;
  t0 u* [, N1 Q! k' Y( Y$ O+ C
Maximum(S) ;
' J; W) ~6 J" m! u& K' DMinimum(Z!a, Z!b) ;
Minimum(R) ;

: U, ~7 B& ~8 B; s& b, V/ ^
1 g' e% R; Y" V/ @2 [0 Z& \
Integer Ring
' e- A- r7 O/ pResidue class ring of integers modulo 12
Residue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
' h. K2 g5 E; \# |: @12
2 a% p4 K2 @5 _5 b12
false
4 \2 D+ T" n7 l9 c: Zfalse
1 |4 K  k( b6 [2 ]( `$ Ptrue
$ w% U7 E+ m- D- z( |# M* @true
true
2 {/ \- U* s3 Etrue
false
true
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
RngIntResElt
RngIntResElt
9
/ \* A7 U+ m2 f- y12
, |, k5 ^; V6 P% o' U0
! L$ V1 I/ W, F0
true
/ r# R) R. T6 U5 Qfalse
; Q7 r* c1 n1 o+ R! A5 j: ofalse
' N9 y  ]( U" u  n" H: Gtrue
  E+ K1 u) {% F% D1 X' w& v& C5 k0 Nfalse
true
false
. Z6 H/ N. c* V$ ufalse
4 e* f% @1 P6 m9 S# pfalse
, b9 ~- l" l* z- B; R; @false
true
true
2 Q% @; M* X& l5 l+ x6 K12
1
3 a& ?1 \3 _+ m& h4 q, m1 W! j
>> Maximum(S) ;
          ^
Runtime error in 'Maximum': Bad argument types
Argument types given: RngIntRes

9

$ \2 E- j- H. }1 ~# M>> Minimum(R) ;
          ^
. y3 o9 K, n0 R+ x% w# d7 [4 oRuntime error in 'Minimum': Bad argument types
' J9 d5 R3 J# O7 C7 m2 qArgument types given: RngIntRes

lilianjie1

7 X, b" w: ^* P) D( k- D
- ]3 |" p( T) A+ r. [1 C* h! yZ:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
- x" O  a8 B7 w$ eI12;
ZZ:=IntegerRing(15) ;ZZ;   
! d( ~; q8 c. E1 z: ?# z- sIZZ15:=ideal< Z | 15 >;
IZZ15;
: d1 N. Q$ B0 r/ b9 \1 c0 n; nI12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
7 u! o6 R6 c% Y7 K: T  XZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
; K) W0 `5 b3 `- D) @$ l: Q. Z6 aIZZZ5;

) J; ~9 g9 s0 i0 o) l4 E2 |I12 *  IZZ15;            理想和/积/并/交，
& B0 v* j% c( F. b# W; o# z理想和是理想对应两（可多个）元素加，
* d! S' b8 b# @+ w: H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，

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;
3 ]& }' l7 Z, B7 u  b! a
) D3 d  H. _$ w, |, M. qI12 * IZZZ5;
I12 + IZZZ5;
, D, y/ Q+ X$ ]4 x8 q* u/ rI12 meet IZZZ5;
8 F7 n) I! T! k5 ?: `I12 / IZZZ5;
IZZZ5/ I12 ;
1 T- }. N" a1 @, ]1 ~9 H/ T7 Y& }Z * IZZZ5;
I12 + IZZZ5;
1 x" C0 l, k" K+ L! ~5 o; w* l- p; EIZZ15 meet IZZZ5;
- y  }0 n% ~6 {4 Q1 R: ?9 z# u8 gIZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
6 b5 e+ `# g: R# tIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;

( n  i( x! y* }' v) a: ~I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
+ n  H/ A+ x1 \  f" R: ~IZZZ5 subset IZZ15;
Integer Ring
+ [, v3 W( _6 \. M! _, J4 HIdeal of Integer Ring generated by 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
8 N- o# a3 j$ |false
Residue class ring of integers modulo 12
, w  {) H- w) ]2 M! U" O$ mResidue class ring of integers modulo 15
2 ^, r( s; U+ O* J, |/ A; w3 V; PIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
Ideal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
0 h. W" V6 q- f! }7 N) Q# Z' gIdeal of Integer Ring generated by 60
* p  Y; y( A6 p+ v) p! w2 V. cInteger Ring
* o  A3 m! M& `Ideal of Integer Ring generated by 60

>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.
' G/ R% x$ ?' |5 @8 D

# F7 x3 \' R9 i2 ]# _. U- d>> IZZZ5/ I12 ;
        ^
Runtime error in '/': Argument 2 must divide argument 1.
5 f4 |/ W* G% p! Z$ F. [
* T$ E: h& ^) OIdeal of Integer Ring generated by 5
Integer Ring
  L6 f0 w2 p) w! h) cIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
( F! z0 o1 t' H5 IMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
/ _+ L& u# i; N! i$ [3 o5 D1 |Ideal of Integer Ring generated by 5
, L2 `- V8 ^- B- }$ K4 T1 K+ RIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
  R7 R% V% C% L& |+ vMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
& ~5 {: X2 y  ]2 }0 k
* l  n, K: C" {+ Rfalse
1 k$ U3 Z8 Q9 g, b& [, |true
true
false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
I12;
, h0 z- W( h" d5 j( U7 PZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
/ S+ B$ ~2 ]: k" O' T7 E$ E/ G5 jIZZ15;
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
- ?7 m' T6 U5 @! @8 G6 ~$ V9 c5 oloc< Z | 19> ;
5 t0 d/ t1 o, N* S2 t) [loc< Z | 17> ;
; r; D5 Z5 v' i! [8 ploc< 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 | > ;
5 l1 o/ i$ b# E) t) o  G6 M- E
& N) {( }/ ^) Q& H7 n3 }# hext< Z, 2 | > ;超越扩张到多元多项式
3 S$ S% f' ^- T) X0 ~
3 P+ e( I, F6 `; c' K9 xext< Z, 3 | > Completion(Z, I12) ;
4 v4 |, S" Z0 L4 {7 ?2 }7 _
/ R" }6 X* U& S) o, }$ p+ f3 g
" Y- x3 a7 P/ f. Y0 k5 C% acomp<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  >;

Integer Ring
5 v3 T6 X. L7 K  _Ideal of Integer Ring generated by 13
& a# w+ j  s* h# ?Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
# H8 [5 \9 R6 iFinite field of size 13
( y4 i% G: i+ `$ S" k' KMapping from: RngInt: Z to GF(13)
modulo 13

: S- U  H' T  ]>> ResidueClassField(IZZ15);
3 ?; H" l: ~, t" D8 U) H  `: `) D                    ^
- A. D. e- n. x' W. l; ^Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes

Valuation ring of Rational Field with generator 19
! {: }' V2 z- C5 Y( uMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
2 m2 d" y, [( A2 ^* sValuation ring of Rational Field with generator 17
Mapping 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
Univariate Polynomial Ring over Integer Ring
6 L  x9 E2 X' J  ?0 W5 z# W* ?5 cUnivariate Polynomial Ring over IntegerRing(60)

>> ext< Z, 2 | > ;
      ^
+ h$ {# q! w. d+ k6 g' D: {Runtime error: This constructer is no longer supported
, l% T$ D( m- W) m  O

>> ext< Z, 3 | >
( h/ X& ^4 F2 W' a# X' s      ^
% y: k- F7 ]5 _' U" LRuntime error: This constructer is no longer supported

13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
) Y3 t6 K9 y3 ~; E: V
Completion(
    Z: Integer Ring,
( S' O6 ^7 j% H7 M    P: Ideal of Integer Ring generated by 0