一些初等函数：

lilianjie

:=IntegerRing() ;Z;
) H& W2 ?8 K9 @( U' Mn := -1666666666234567890;
/ H) u2 }, \/ V% C3 k> n;

> n:Hex;                           转16
5 o) v1 n& R4 FIntegerToString(n, 2);        转2
. b, p( k, q5 |' p8 k7 A% HIntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
( W( a3 v0 D% G9 R+ n; IIntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
Representative(Z);         环代表元
$ j9 U% ~- x5 m9 x, L9 y* `Eltseq(n);                        取整
# S+ ]  b0 |& \' Q$ [Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
5 T+ U8 @/ f; ^0 ]: s+ v/ l
9 M6 p( a+ I8 ]5 `$ c6 Z) \" E# Im := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
- l- o# J6 F' {- u: @' H$ gk := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
n eq k;
9 Y1 w5 F: f) c/ i: \& c/ skk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
4 X1 o# {! `. B0 Hkk eq k;

( g# K4 _4 ?) \9 q  l7 u: jk := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
! b7 F$ T5 d0 Cn eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
0 l: a) `" I9 O  K) [> kk;
+ B0 s! a3 x6 n8 `& Tkk eq k;

Eltseq(kk) ;Eltseq(-1/14);
$ S) G2 Z  p, v( W
2 L+ F% s* D9 T6 V6 {2 y9 q5 W7 ]

. L" U+ D2 M7 A( G+ k
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
* Z" o' Y3 T* i& g7 j: j; g3 v7 R$ yn eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
% L# E. n% Y* F2 K5 J& jkk eq k;
7 d5 c8 O3 F$ ~: @, x: T8 y3 y
k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
2 x6 p+ k( n6 x; p> kk;
kk eq k;

- c0 T, f2 [; E& |4 O$ u8 ?Eltseq(kk) ;Eltseq(-1/14);



) D: x& W" o! o
4 O0 V8 o& J# @' E
% y1 [: {: j9 E* h5 O; f
3 Q# S  Y3 i4 @& Q

( f6 q% Z7 i& w/ V$ e3 C7 y8 W5 h

' _1 X  r" }$ _  b" t% u( b=============

  [( S5 F# O6 ]$ f, i: x% k1 L
1 {2 S9 z/ H9 r$ k& F
% }. x, }1 ?% A( c7 v" P
8 O& d$ h; N. K: k7 f  BInteger Ring
. X2 [. U3 W4 e2 X-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
3 B& G$ w7 U3 c( j* h# g-CNUO0WGPY9CI
% v8 V* h6 b) P2 r1 L) w7 t-1666666666234567890
: K+ \9 H; o& s- w( ]; S# w4 o-1666666666234567890
8 }# W6 K2 Z# g  a5 d0
1
. U0 X4 W& i5 J3 }0
[ -1666666666234567890 ]
[ -1666666666234567890 ]
2 g; d8 s" L; I  }' o: [1
13
7 b' k7 n/ V/ m% k$ H1
2 r4 C! ?' b  K7 n9 g4 g$ O% x
-1666666666234567890
) c  }- t# k; ^6 ctrue
0 q( @/ ^6 f+ G3 \-1666666666234567890
true
+ ~7 W! O0 k  D5 K# M3 k  ~- Q-1666666666234567890
true
-1666666666234567890
true
[ -1666666666234567890 ]
[ -1/14 ]

0 S0 G$ Z' q6 w5 o' R4 k; X4 M# b

. D) i8 e! I' o; D

5 F, Q! T8 t+ ^" W# B+ E5 y; V  Y

-1666666666234567890
-1666666666234567890
+ w" w" L! }4 ?5 x* ntrue
-1666666666234567890
- j+ r. x5 e: Mtrue
-1666666666234567890
* \+ e: {; h1 }( [: ]9 X7 G, A/ \true
; _1 P! S" b9 |6 R1 u8 G-1666666666234567890
true
% R# Y) |6 f) y  W# J; P* T[ -1666666666234567890 ]
[ -1/14 ]
, ^- u) \+ z5 Z$ [1 {3 g

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
/ A- g. `- s; d4 |% n- L' Is:=0x12345678111;ss;
) N# P2 F1 t6 V! M+ w! h
0 b/ U. K9 p2 T# N# X, Nsss:=Factorization(ss);sss;
) R* p0 e9 ?0 G  s' }' I# Z* s1 b: Msss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
" J4 B% V  J3 ^7 {, y: T' T# pFactorisationToInteger(sss1) ;
- k5 I$ W7 e5 H" a6 ~- \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);
* N. z- g1 v2 O) v7 H' qssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制
1 a( F. j" v' ]/ O& ~
$ x& n; x$ J  n3 y; T12345678111
12345678111
, c+ F( @: N( w9 Q6 \+ E7 c[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
4 I+ y: k7 [. A, ~) R# ^$ E1250999894289
" K7 \7 \* E. }3 V" G$ }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,
+ R. _3 w, q$ ]2 ]/ I# r1, 1, 1, 0, 1, 1, 0, 1 ]
8 R& f& V4 J$ b4 C1 j12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
1 \- V. `& ^) y) T/ B( Y12345678111
( Q; s9 v" t* _& f/ I- O0 D9 J5 I[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

lilianjie

Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
) z; s2 r( b/ v" @# M' X> n;
n1:=Z!1111111111111111111111;n1;
+ l: R" B7 g9 c! un2:=Z!11333331111111111111111;n2;
( I, |) P& ]3 u0 F: _! \7 _
+ S: t- D" s4 J& |) z+ v! ^" K0 u
' u3 W% _: ?% c. E8 ^% YK:=Z!n1+Z!n2;K;
% b  H) X- G5 ]5 q& H
IsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
5 X9 L) B2 x5 MIsCommutative(Z);可换吗
3 h4 ^/ l4 t: u: 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。。。。
  b  A  p' `( s2 M9 D5 Z8 `/ bIsPID(Z) ;主理想整环吗
% ]3 p- u' b- C& j  v3 S3 D, R$ w
+ x; ]/ m% q- }0 [% r# ]IsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(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
% ^6 q2 \6 b, e# y0 uMultiplicativeGroup(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
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

: D7 {- L) P; v1 u# JZZ:=IntegerRing() ;ZZ;
+ x2 Z7 ?9 r; t1 Z6 [ClassGroup(ZZ) ;
; t1 ?6 L# N8 N3 i, Q: p& m
===========

Residue class ring of integers modulo 5
1666666666234567890
1
7 |8 y) i7 S1 u2 d1
2
: h- R/ t& E/ C, S6 u5 {true
2 q/ a  @9 x& c7 ?* }$ c5
1 T; q- Q8 z# ~& \) Qtrue 5
( [) M1 H; _3 M, v9 C+ itrue
' O2 |) U% H8 F& r3 Bfalse
: j3 A- F( z' ?$ `; x* F) x0 s! ctrue
true
; y6 o# y! x  N  q5 v/ \true
true
true
% E2 c; V1 G7 z; d0 q- M  }true
! \( v! B' h! wtrue
6 m* |- h8 I! r9 Z8 A$ S2 WResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
" ]/ ^+ M4 f8 P2 |+ @Defined on 1 generator
Relations:
5 J2 i3 e  U' d. R3 U    4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
2 h4 Y% S: P7 L6 Q- XRngIntRes
$ L% i* s, E! b5 i% R( l& P2 e" DPower Structure of RngIntRes
* l( ], O6 z3 d5 ^# `" x6 BResidue class ring of integers modulo 5
Residue class ring of integers modulo 5
2 Q4 v0 P* W9 A  n& J, ~3 S0 lAbelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
    5*$.1 = 0

2 @% Q; B, y! {; U$ B% `! F>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes
9 ~& N: O. R/ o, o. N/ `
Integer Ring
4 e& `6 ?  F: d1 [Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
MultiplicativeGroup(Z);
3 ]/ `, w6 ?5 _9 TCategory(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;

Z:=IntegerRing(13) ;Z;   
% F: x4 G0 y; V4 m2 U5 c& f3 B0 ] UnitGroup(Z);
5 I$ {/ z0 M3 t* |" F! C! eMultiplicativeGroup(Z);
  F4 }) N; @0 F4 r; nCategory(Z) ;
PrimeRing(Z);
$ p3 E8 q/ [# T5 S# |' g0 `+ oAdditiveGroup(Z) ;


$ s5 ]* O$ X$ U% t2 n
9 v5 O* I) l  l+ o! a9 u; @; QResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
5 w6 |3 O, H( O, `& T; \) d4 FDefined on 2 generators
' N/ [" t/ r! W3 S+ ^9 c; W& {) bRelations:
: n  X% Z7 _: s( n, ^0 Q    2*$.1 = 0
    2*$.2 = 0
1 s  S. a6 E& hAbelian 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
" A& O# T  N) H( y+ }& H0 u& sRelations:    2*$.1 = 0
3 O- ~* G0 R; f1 A    2*$.2 = 0
5 \, ?. k0 O6 x7 ~" LRngIntRes
Residue class ring of integers modulo 12
) k" a- e) }, T0 f/ h% C0 Y4 ~( wAbelian Group isomorphic to Z/12
: [5 }0 u' f( ^* F, l+ FDefined on 1 generator
Relations:
    12*$.1 = 0
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
  U! r* J- |2 x4 u; WDefined on 1 generator
Relations:
    12*$.1 = 0
3 `/ ^6 s: }, a, J! hAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
- H# Y4 s6 ^) T9 sRelations:
3 ^* E5 Y0 h0 [; p8 _2 i    12*$.1 = 0
RngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
Defined on 1 generator
$ Q' m% g, k$ ^+ y: K0 X: o1 @; [$ wRelations:
    13*$.1 = 0

lilianjie

' O7 p4 E  k8 G6 v3 d+ h
Z:=IntegerRing() ;Z;   

R:=IntegerRing(12) ;R;   
. j4 v% ?- n! D( u+ P- |S:=IntegerRing(13) ;S;   
0 q" y1 V% z0 i1 J
4 J9 g& j. Z8 w0 T: P: E
PrimeRing(R) ;
$ D0 H# Z; [" ?. G. yCentre(R) ;

Characteristic(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) ;
Has**(R) ;

IsPID(S) ;
: U& U, N; p% E1 l8 AIsDomain(S) ;
Has**(S) ;
' G! u  Z' z/ q3 O( OR eq S ;
R ne S ;

Parent(R!123) arent(S!123) ;
. P, C$ x, K5 s8 G7 M9 f# [/ lCategory(R!234) ;Category(S!234) ;
* j9 B- Y6 K/ y+ L: d
a:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
Representative(S) ;

(R!a) in R ;
(S!b) notin S ;
IsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
! r+ X9 u% u- Q( VIsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

Z!a gt Z!b ;
Z!a ge Z!b ;
+ l/ ~; P; x. X( k( d8 |7 g! P8 g; w( xZ!a lt Z!b ;
, [7 {0 x. k( j  yZ!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
2 ?+ I3 p# w1 B7 tMinimum(Z) ;

Maximum(S) ;
# D7 i, V. m$ aMinimum(Z!a, Z!b) ;
0 Z, {" ]$ ~3 t- z; S% q* a" c" VMinimum(R) ;
: d) }8 ]5 k: n6 G3 m# W' f' C$ h
- ^$ k! Z  x! N) z: [* J
7 U# c2 \- H+ D6 L
Integer Ring
: e: ~# _: b0 A* hResidue class ring of integers modulo 12
& B- j6 C  ?+ X- Y1 z+ ]Residue class ring of integers modulo 13
# |7 M. {; Y" q$ w" }& C4 \, NResidue class ring of integers modulo 12
Residue class ring of integers modulo 12
, H3 T2 e) p" Z% w9 m7 T& z12
12
" A2 |+ k" X9 P1 L& @false
false
true
! k0 N% ^: d% F' M0 h% u- P! ztrue
+ A- P1 k7 y2 P% q4 i" a6 gtrue
1 }1 g: D1 H. [+ _8 r& jtrue
/ m# @" T1 H# r+ J( V  sfalse
true
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
RngIntResElt
) R0 F/ k  V9 u5 r5 V  B! zRngIntResElt
* t$ k0 {( V5 A9
2 }! k5 G  Z( z1 C8 @2 c; ]12
0
7 f& z( N4 L- `0
0 G! U5 [' v, p& [" e; h# Vtrue
false
! m$ G7 C' u/ }/ o  W8 yfalse
true
: F) \6 Z- E" c$ _6 c0 `0 Ifalse
true
$ X2 t6 }/ Z3 cfalse
7 P( l7 s+ h9 q" D2 ?false
false
- h$ X$ \7 I9 K3 P. K" h# dfalse
9 ?3 z- g) q; [; mtrue
) ]8 [+ I1 k* K' i% Strue
12
1

4 Q1 e0 s8 _. @+ s>> Maximum(S) ;
          ^
Runtime error in 'Maximum': Bad argument types
Argument types given: RngIntRes

& T. H* D5 Z  p2 Q9 C+ S( M# y9

: r6 Y1 Y( i" w4 T>> Minimum(R) ;
          ^
$ L; K5 `( }# l4 u  f8 y* q/ [( pRuntime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
* ~) V+ E1 V. b' A- T; M, AZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
IZZ15;
6 e5 `5 E0 @8 F( t6 W; J3 SI12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
' B+ X) B4 h2 vZZZ:=IntegerRing(5) ;ZZ;   
7 k/ s0 R' W& qIZZZ5:=ideal< Z | 5 >;
2 E* d# ]  \) S* }9 ~IZZZ5;
% w% H' g/ B) z! @' o& x* Q
$ Z. i. V6 m) ZI12 *  IZZ15;            理想和/积/并/交，
* i" p1 _9 @5 b; z. l! \理想和是理想对应两（可多个）元素加，
, J) d  {" Z3 ?理想积是两理想（可多个）对应元素积，
( |4 r( F9 l, L; e5 z: O4 R理想并就两（可多个）理想元素并，就不一定还是理想，
8 u2 @2 [0 F& F- N: X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
4 g  @& k+ _+ ~I12 +  IZZ15;
I12 meet  IZZ15;
) j$ `# @, s2 G- O3 h
4 f4 m  D7 K% B& Z) o5 v8 o; q- NI12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
, S2 c+ p3 Y" f# t" nI12 / IZZZ5;
  O! D+ `7 {* T/ [5 f8 o- NIZZZ5/ I12 ;
Z * IZZZ5;
/ ]2 @5 S5 E  x* ]# vI12 + IZZZ5;
7 z. Z3 U  B& _0 _" fIZZ15 meet IZZZ5;
4 o. V, S. m1 j" w3 ~IZZ15 / IZZZ5;Z meet IZZZ5;
* u$ T) r) g( }  ~5 D% aI12 meet IZZZ5;
- V8 p! `8 Z' h5 |IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;
- ~4 l& w" O/ p# A1 {# w8 k7 ?' S
2 I+ Y1 i4 g0 l/ ~2 q7 `5 dI12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
IZZZ5 subset IZZ15;
2 n3 D" [& l0 h$ n3 z7 HInteger Ring
0 B+ A( g# g$ P1 K7 i) B2 U2 c: t- \Ideal of Integer Ring generated by 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
2 p4 d  x6 {5 M2 Vfalse
Residue class ring of integers modulo 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
0 i. O, d$ L0 X7 j7 W& lIdeal of Integer Ring generated by 180
$ I) {: i( ]& h3 z8 tIdeal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
, Q9 d8 d- Z( i; c: d$ G) A: s* T( LIdeal of Integer Ring generated by 60
Integer Ring
' J/ d8 V! T6 |7 {* h- @& C$ GIdeal of Integer Ring generated by 60

2 t% @+ @* ?  F0 P9 x/ P>> I12 / IZZZ5;
& q9 ^6 P. Y+ W  i* D. L       ^
/ P7 ]$ D5 L" p4 VRuntime error in '/': Argument 2 must divide argument 1.


>> IZZZ5/ I12 ;
5 [( E( ?0 d! b9 ]. F% E4 Q        ^
Runtime error in '/': Argument 2 must divide argument 1.
! B- k/ W" {! P) R$ v
9 s5 a: D4 R. KIdeal of Integer Ring generated by 5
Integer Ring
Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
* F# a* z9 K7 G: I$ nMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
/ T* c4 z3 G8 r8 zIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
. x) R% V4 t; w# L" E. GMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z

false
. v' A7 {5 K0 ]. ]true
6 g: a. r  z' y: `; O5 o3 jtrue
false

lilianjie1

Z:=IntegerRing() ;Z;   
& @2 r% k; Q# w( e, DI12:=ideal< Z | 13 >;
. k8 d; {$ [8 N$ QI12;
ZZ:=IntegerRing(60) ;ZZ;   
! W  s! ?2 m$ X8 |  V) L6 c& A" PIZZ15:=ideal< ZZ | 31 >;
2 X& O0 q* I! [5 i- `0 u& v1 @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
; D3 w+ ?& U  ]3 A. A" P9 q) n+ dloc< Z | 19> ;
7 `% o, ?$ J. S" ]6 M+ g- ^" bloc< Z | 17> ;
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 | > ;

ext< Z, 2 | > ;超越扩张到多元多项式
: y2 [, k  Z* j! ]
7 d8 h3 L! F2 X: aext< Z, 3 | > Completion(Z, I12) ;
1 v0 P3 f. l7 b5 W- I, X) r
+ _- o$ E" Z; H3 a; O* k
; ?" M( ~5 Q# xcomp<Z |I12  >;
    素理想零理想完备化，和P进环联系起来
* A2 R, ?# k3 H, lCompletion(Z, 0) ;
! ?- }% [" n" U$ Qcomp<Z |0  >;

4 ~3 q) _: e" ]' P) a% u5 nInteger Ring
9 k# T, ^$ n: k1 {* V3 OIdeal 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
Mapping from: RngInt: Z to GF(13)
modulo 13

>> ResidueClassField(IZZ15);
                    ^
Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes
' R- H% M5 G7 |8 o! ^
Valuation ring of Rational Field with generator 19
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
Valuation ring of Rational Field with generator 17
3 k7 m  Z# c; X+ {  @$ NMapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
3 k9 O( O4 m: F# iValuation ring of Rational Field with generator 131
- Z' d: g9 U; L" RMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
, m1 V  ]1 L" \1 P3 Y5 QUnivariate Polynomial Ring over IntegerRing(60)

  F; g" F1 ~4 S9 `5 @; O>> ext< Z, 2 | > ;
      ^
4 t. |7 D; A. ~1 G; K) PRuntime error: This constructer is no longer supported


>> ext< Z, 3 | >
      ^
% D2 M- q5 Z. {- [: N# ORuntime error: This constructer is no longer supported
+ E  ~9 y( ?" {) M5 g1 i
" j9 s/ R# i% D0 n; s& \* m13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)

Completion(
& d# J) ^' w) Y1 @7 u, l    Z: Integer Ring,
( _8 m2 |% a1 ^! ]+ m+ t& ^    P: Ideal of Integer Ring generated by 0