一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
5 w& B+ Y( v) q6 n# S; M; s* E> n;
9 F& Q# O$ a- f3 ?" {2 m
> n:Hex;                           转16
IntegerToString(n, 2);        转2
8 w' A2 L8 e& u% g, H' J, E4 P( I. zIntegerToString(n, 10);       转10
$ L( J; [  R' {8 _% }! z) bIntegerToString(n, 16);        转16
1 |: z2 U% y# p) l3 ^IntegerToString(n, 36);         转36IntegerToString(n) ;
4 a8 d8 ^, D. VIntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
: G. k* W4 {' r6 }' w: U2 JRepresentative(Z);         环代表元
Eltseq(n);                        取整
( G% v& z% v7 XEltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);

& F5 }0 A- n* c0 G0 u& y% }1 im := 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;
* S$ @' q$ X  Z$ M$ l- ]) U' rn eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
" d$ w6 u) E1 j0 X5 p; f> kk;
! D; T" X: P! j3 q! X! \9 u- mkk eq k;
; I4 _, B: q* @9 z5 l. D
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
0 ^& _: b$ n* U  A, Ikk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
: |/ U: m3 s; F3 W  ?# D# n6 K
Eltseq(kk) ;Eltseq(-1/14);
8 @% P9 k" _  F; {



7 B! `) V2 k6 [1 S' U: \  n7 Nk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
- n' W( N  `9 D; Z3 W+ k2 M3 c3 a8 en eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
7 M+ r* c! y3 A9 r0 U> kk;
kk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
0 t3 c. n' }% k0 y3 l7 H6 V+ F$ R, \> k;
n eq k;
' p7 a; K7 j; C  Dkk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
9 E& h. O5 I. l9 T7 Z# b  v/ O> kk;
2 y! J+ K; z2 B0 N6 T5 Qkk eq k;

Eltseq(kk) ;Eltseq(-1/14);
4 |" s: a8 o3 y+ ^, ^* s* R


% c6 c" }" `, K$ G1 ]4 @" D; G




9 {: W- t: M2 t! G. l: h
4 q" ?0 G- M8 a( J
=============
; ~- G6 s3 V) n6 @* R

( P" l% L3 F0 A1 B; J+ d2 n
) [0 n2 T/ B5 f& I. O
/ i' G; H  p0 tInteger Ring
-1666666666234567890
  v" G- M& Z4 v* O5 i& }) f-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
% e- @! P7 Z  S' l  s9 N- p-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
0 R& _5 q- X' m( m% E  h+ j; Y-1666666666234567890
( M* r7 @1 T+ V7 c% U0
' u# |9 b. C, f) M" `4 J1
& v# D9 a+ _, i& o8 f/ ~. w- H9 B3 U0
[ -1666666666234567890 ]
[ -1666666666234567890 ]
& ^4 g6 {" z5 k, b% r6 J1
  Q7 h/ c; c; D$ i/ x13
1
8 U5 @9 E! K. W4 l- E' ^
-1666666666234567890
) L/ \" U1 r( j  ctrue
-1666666666234567890
true
+ W' v, u, \3 S" H3 [0 O3 C2 w-1666666666234567890
true
) B0 x$ _6 ^% Y/ J4 s-1666666666234567890
8 b  R: @+ v) L# z, X; atrue
; }" o) V+ v8 p8 Q[ -1666666666234567890 ]
2 n+ u1 Z# [/ y[ -1/14 ]
( Q3 n$ D* L+ A) G3 F. }0 h

* m1 p% s% k- X9 H. q; W
9 T  x; D4 ?9 E4 N1 x& X, @
3 u5 W' l! }% t8 F! N# H$ T
( G9 c$ w2 o. c7 o$ [' m; D# j2 a
4 l7 @  A( q* J  F3 k6 c4 |
# U8 B$ @7 }/ r1 n-1666666666234567890
* E/ B- K5 q# Q) Q. R1 k-1666666666234567890
' ~  g* L2 j: R! M. K" ?0 Dtrue
8 N' u* _& h9 f; U) t0 i+ H( ]) L' b-1666666666234567890
) x4 U# y2 O% l" s8 `3 l' F, Vtrue
-1666666666234567890
6 N% e* l! m/ ftrue
, l( e3 W1 N( R  }  B4 c. D6 O-1666666666234567890
true
[ -1666666666234567890 ]
[ -1/14 ]

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
" l& t! M% m3 A. w8 p! i, xs:=0x12345678111;ss;
; A. v, P( f5 @9 C! e
' V  O4 w6 L$ X. {sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
5 Y: E' ~" C+ z5 c( YFactorisationToInteger(sss1) ;
5 o* }3 o- R" E2 \* i' q6 vFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
% x2 t! O7 }; u% ?  u* X9 }+ ]9 v6 Vssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
! U0 C8 L1 J% k- ]# R- u! k3 I1 Qssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制

8 @- K" N. f. [7 `, e( o6 n12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
% u5 B& A, n! S% R0 A) U1250999894289
4 I$ \4 F' f4 ~8 r& _7 D  O2 L[ 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
, V+ B/ b( p8 X1 |. v; m1 @[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

lilianjie

* ?. E5 d0 W, e6 ?, U* V
9 u- _0 x$ |( }& y+ d" F7 r4 KZ:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
6 Y+ H3 S& p+ j0 B/ ^> n;
2 s( s! Z6 ?5 V' |) vn1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
; B: x- k& ~% I
4 e- ^' n9 J( d& P7 I
K:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
7 |: j1 z; p8 A- g/ U# O8 `IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
" }: O  r) e1 t6 a/ Q+ W2 N' I3 xIsOrdered(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

- b* I; A/ |* x" yIsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
2 S6 O; n! k) T9 s% x" q7 @IsPrincipalIdealRing(Z) ;主理想整环吗
IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
  G! V# j& A2 c4 o/ x+ JMultiplicativeGroup(Z);乘群
8 c+ U( V# R2 Y* Y# H* H3 zCategory(Z) ;范畴Parent(Z) ;父环
- i1 w, ]3 Q1 k  U5 xPrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
( ]* @8 Q# S5 S# ?+ sAdditiveGroup(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

' \5 H' P+ W# D; v3 `ZZ:=IntegerRing() ;ZZ;
3 K0 y4 g. q- c& YClassGroup(ZZ) ;

===========
, d& Z% |# Z  s. T( y
3 q" L! a6 U( e/ U* p3 w; @0 nResidue class ring of integers modulo 5
1666666666234567890
1
1
2
5 M3 U5 H) e, M- c5 ]; Ltrue
5
true 5
true
* F+ v+ K) ]8 C* v) Vfalse
true
true
true
1 Q- e7 T; |; g% t& Vtrue
" ?+ @9 j2 D2 Mtrue
' F6 H8 D; n7 G5 \) J- w# O8 q! H& |true
true
+ X9 S- t1 A0 \& \5 ^& _+ ^& oResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
# I, t+ k$ Q$ T- B; d5 B- b" sDefined on 1 generator
Relations:
8 [9 E" X; @& B$ f9 h* e    4*$.1 = 0
  t& M& a: S! e! |( y; `Abelian Group isomorphic to Z/4
& g9 D4 l, J) c0 U/ k! r7 m- i8 dDefined on 1 generator
Relations:
6 ]3 s  x5 `5 ?    4*$.1 = 0
RngIntRes
5 t- H! g' J, U: m2 f6 |2 KPower Structure of RngIntRes
: D) d% B0 a# b5 kResidue class ring of integers modulo 5
$ V; j* b: m4 L  ~Residue class ring of integers modulo 5
/ Q7 L( Q0 W1 v& W6 S  ~3 HAbelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
    5*$.1 = 0

. D1 Q1 L% a+ c6 Y>> ClassGroup(Z) ;
             ^
0 \- g- o0 i6 r7 P2 L6 L3 eRuntime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

1 A$ A. Z% ~4 JInteger Ring
6 g8 c, T2 V2 r$ g# [Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
* ^4 e) `- B5 l! ? UnitGroup(Z);
9 i  m# ~) h6 g: JMultiplicativeGroup(Z);
. j0 ^. u. B) ~3 a+ O+ f5 ZCategory(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;

Z:=IntegerRing(13) ;Z;   
  y4 ?' G. v: v  ^' o UnitGroup(Z);
MultiplicativeGroup(Z);
# J, T; B  p6 ?Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
. G" P, z9 n& g% ]
$ n) `$ p: [/ Z' {3 e7 X$ c
+ z5 p1 o0 R; B" |: V2 _  z
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
# k6 m& g' N0 _4 |" J& Q! dDefined on 2 generators
Relations:
    2*$.1 = 0
    2*$.2 = 0
, o! ]% {' K7 B; |% tAbelian 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
+ X& V3 {: c. W" Y7 ?# F; r" L/ @/ NRelations:    2*$.1 = 0
    2*$.2 = 0
) f1 d: v9 o% Q* l8 hRngIntRes
5 C. Y" r* ]! v& C# p; G# SResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
    12*$.1 = 0
2 \$ q4 H- h1 X' C0 ]/ ?Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
3 `/ h: [# D3 s5 W5 `, X; r  gDefined on 1 generator
- ^7 [3 U/ ~6 e/ K( S+ q) `8 i7 \2 K; pRelations:
    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
    12*$.1 = 0
RngIntRes
' E. [3 e: {0 n- ?3 NResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
  A# C& J/ o& v5 jDefined on 1 generator
* }4 e, ]5 v; uRelations:
    13*$.1 = 0

lilianjie

0 h) I( A6 Z: g6 {! a# @7 p
Z:=IntegerRing() ;Z;   

! m& O+ P! ?8 I" n' S1 v% mR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   


PrimeRing(R) ;
% r& Q% Y$ j* Y/ G% U+ e7 f- @8 {9 t" `Centre(R) ;

0 V: a& Q# V: @1 A# uCharacteristic(R) ;
# R ;阶----元素数
0 G4 ]' s! E* f5 B3 O2 LIsPID(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) ;
( t$ L8 `) V) f( NHas**(R) ;

IsPID(S) ;
IsDomain(S) ;
+ |, z5 S8 e* q* DHas**(S) ;
# {0 A- X' P# X9 l& i4 w0 z. GR eq S ;
R ne S ;

Parent(R!123) arent(S!123) ;
4 ]5 M) R3 l1 d- f$ JCategory(R!234) ;Category(S!234) ;
8 q! l9 e  G0 E7 ]  X" c
a:=Random(R) ;a;b:=Random(S) ;b;
& z$ E; `" s# Z1 aRepresentative(R) ;
Representative(S) ;
; A3 X& u+ o( Z' X1 B6 e/ ^0 D
(R!a) in R ;
(S!b) notin S ;
$ @0 d0 a2 }7 L$ g0 A8 hIsUnit(a) ;                是单位吗
; o9 ~; z+ j. \0 y  e3 pIsIdempotent(a) ;是幂等元吗
IsNilpotent(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 ;
9 ^5 z8 P0 c( ]4 A! GZ!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) ;
) V8 c- ^" s4 c' X( iMinimum(Z) ;

+ f- M) B7 ^* b) AMaximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;
& I- X& P& {4 K% h1 {' _6 k& A
3 C7 r; c; k. ]3 }& ^

Integer Ring
* E$ _. z+ b3 j5 \9 V/ gResidue class ring of integers modulo 12
5 {2 s7 L4 k8 p$ I8 g" QResidue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
12
12
false
4 J# c1 |7 a8 B3 M! Zfalse
true
true
9 O" z! ]- x+ m% H- Qtrue
6 `$ |2 S7 n: b" ztrue
false
% E# n- }4 t) N# R+ f1 ?true
: g4 W% J+ G& u% n4 T( S* UResidue class ring of integers modulo 12
Residue class ring of integers modulo 13
RngIntResElt
, U, f3 f' K* y) @  B( r$ nRngIntResElt
9
12
' [/ ?1 j4 ?/ L2 W; C& h0
+ {6 S; N0 O+ g) f/ ?2 t0
true
false
- \9 C* A. A- G8 `0 b& Vfalse
3 W5 p- |8 w( ]8 ntrue
false
/ x3 [( o/ q6 K8 jtrue
# d% D% v+ ]' Y' ~false
false
/ R3 }8 X5 ~. H' ?( T4 y. S) yfalse
3 L' }* f6 ]- V) `" s; y0 G$ hfalse
true
/ ?# `* o( ]; ?6 W, Ttrue
! n9 N4 C5 }8 @" X* C12
- z' z* c0 f4 G, z4 a1

>> Maximum(S) ;
* J: l$ C& s3 X  J- m/ n          ^
Runtime error in 'Maximum': Bad argument types
Argument types given: RngIntRes
, b; I- M* n" f+ U' L' w. y6 m
2 b) O. X# V! o& P( Z9
- _% W, c) W" N# n4 \$ a
+ m) k" C6 y/ V* S. \>> Minimum(R) ;
! W1 z! M% C0 D5 n! E          ^
/ |7 x! b6 e0 ^% g( TRuntime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

lilianjie1

1 F8 O$ ~/ S5 KZ:=IntegerRing() ;Z;   
( [  S* `+ U1 }, G" BI12:=ideal< Z | 12 >;
1 V$ N! f* C2 J0 \3 p1 GI12;
ZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
ZZZ:=IntegerRing(5) ;ZZ;   
/ P) p& a& K2 x9 ]. Q% GIZZZ5:=ideal< Z | 5 >;
IZZZ5;
( `; f/ g. f2 S5 C9 Z
I12 *  IZZ15;            理想和/积/并/交，
3 f; r6 w0 K7 _理想和是理想对应两（可多个）元素加，
! g& A$ L  n/ X0 t7 l8 C) 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，
5 H9 t, W8 y9 b/ I理想交是理想（可多个）元素交，理想交一定还是理想，
* m$ V. u- {$ e2 U; r
) d" f% n5 _" \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
+ G: J7 Y- k5 A1 A% vI12 +  IZZ15;
I12 meet  IZZ15;

I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
) S; ^! v- ~: u$ {: G6 y: d" cI12 / IZZZ5;
5 _" c8 {5 e8 R0 sIZZZ5/ I12 ;
3 G' k4 L* G' G- lZ * IZZZ5;
- K$ |0 y8 O# m! S" o2 q( d% K' G: gI12 + IZZZ5;
IZZ15 meet IZZZ5;
5 C0 l; O) e+ T( V, GIZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;
3 Z, M6 e' f+ U$ m# [
+ u% s! M6 @  nI12  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;
) P2 {8 [3 T' `$ j. B$ m4 ZInteger Ring
8 O4 h2 U4 l) I7 T( e6 ?. yIdeal of Integer Ring generated by 12
! E* Z) b& W8 m6 s" iResidue class ring of integers modulo 15
8 T8 N- i5 [* `. b  ~Ideal of Integer Ring generated by 15
7 v3 X- J- W$ E0 S/ rfalse
' B# @6 ]& R$ [, j5 z$ GResidue class ring of integers modulo 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
) D% J3 k1 A& AIdeal of Integer Ring generated by 180
( B5 M5 F2 _* XIdeal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
Integer Ring
) G4 Y9 W7 A) z, @% P3 w1 y* ~Ideal of Integer Ring generated by 60
, `. n" R5 n# {! R6 V# J. l) T8 I
0 \& L7 x  Q* }( d1 S4 g+ y( w>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.
. ]# E) \2 s5 K# u7 R) l5 O0 p

>> IZZZ5/ I12 ;
# q4 }" x1 D9 x+ ^2 e        ^
0 u! X$ b* T; N! mRuntime error in '/': Argument 2 must divide argument 1.

Ideal of Integer Ring generated by 5
Integer Ring
4 d6 V9 a. j- X, CIdeal 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
Ideal of Integer Ring generated by 5
( b5 M$ M- q6 d+ Y  q4 U) K3 X0 fIdeal of Integer Ring generated by 60
$ y2 _" O% }/ X& y6 R( N# uIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
5 S' ~: P/ [9 u% R1 PMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
4 ^2 o2 i$ I4 ]* z
9 k3 ~# E  T# ?$ b( h5 h2 |false
true
true
/ C$ \( ]$ R# Q1 Y5 \false

lilianjie1

Z:=IntegerRing() ;Z;   
+ R3 i  a' ?' A( v2 v. @# A+ {I12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
IZZ15;
ResidueClassField(I12);
: b5 |' U- ?7 a. e, A1 ?" J* i& 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
loc< Z | 19> ;
6 H* f& O7 ~' x* G9 x. J' `/ I' ^1 gloc< Z | 17> ;
& B, `) y" O+ q8 t' I5 C3 ~  e: {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
5 C  ]% z. w+ f8 b3 W/ j5 Q6 q) gext< ZZ | > ;
! w' I3 b; y: H3 }2 q
  R3 r+ I- C0 R, Wext< Z, 2 | > ;超越扩张到多元多项式

' k9 W, d# F1 m5 H4 \$ R6 b1 Yext< Z, 3 | > Completion(Z, I12) ;


/ @  ?- e' Q) Q. v# {3 [comp<Z |I12  >;
    素理想零理想完备化，和P进环联系起来
5 O) k9 V2 ]% s9 g9 z- X7 j6 PCompletion(Z, 0) ;
comp<Z |0  >;
/ d, e. c8 t9 H1 q* U) i( V
( B$ o% Q) w& h6 \  t( M% LInteger Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
' |/ V4 T7 K6 g2 U) Y9 Y$ lFinite field of size 13
2 Y, H% ^1 Y3 x9 VMapping from: RngInt: Z to GF(13)
modulo 13
( P% [3 E5 a$ r: R1 @2 V
>> ResidueClassField(IZZ15);
                    ^
0 A9 s7 C  `  T1 ~) VRuntime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes
" _/ l1 z+ @; D5 Y0 A. `. K1 n
; _8 b# {8 B9 K/ i/ LValuation ring of Rational Field with generator 19
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
+ G% [, J# y  I$ A2 gValuation 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
& J7 N0 q8 i9 HMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
/ d* b, }3 c5 v7 F6 KUnivariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)
, Z* N0 i+ h- A
) |+ K, I) q" W5 p8 |>> ext< Z, 2 | > ;
      ^
Runtime error: This constructer is no longer supported
& \4 }$ A; L# i

. S# w) S- e1 a  O6 l* g>> ext< Z, 3 | >
      ^
. m( H& ?1 a4 c$ ^9 c' uRuntime error: This constructer is no longer supported

13-adic ring
/ J3 D6 x% [& k# SMapping from: RngInt: Z to pAdicRing(13)

  E( _8 J1 t' G0 u  ?+ F: O+ `7 W  DCompletion(
4 d+ l* @& W- z! z    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0
/ D3 S2 E, C/ S7 o+ }( n# P( I% V