一些初等函数：

lilianjie

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

> n:Hex;                           转16
' U7 l  c% G3 S9 `IntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
0 @4 k  e8 Y2 e' z. AIntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
- w$ S& M0 [: ^IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
, t5 y9 w2 }5 b/ a2 ~8 gIdentity(Z);                  
Representative(Z);         环代表元
7 N4 i  e1 k# A" eEltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
: H9 O- `6 g; d0 J6 c  S- ]9 f! z
+ u! ?* q  N8 ~& _  i9 em := 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;
. A- G% [) o/ \; Z) e9 W. {& O* pn eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
2 C( j/ W& n4 G9 Z: H* {' g3 w> kk;
5 U! d2 p' j/ [kk eq k;
/ n5 |! p- z! }4 `. R/ k
0 w- c! R3 Y3 R# u' Y8 R' [6 wk := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
+ h9 b! i% Q! q% V( p> k;
n eq k;
$ m/ Y  P0 V- g  w3 Hkk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
' K* j, ?4 n+ Okk eq k;
0 i8 i" K7 P9 q7 m' O) e
Eltseq(kk) ;Eltseq(-1/14);
: v1 b" q$ p& R. c


9 H# ~/ C, J. S
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
9 }( s; _- h4 t$ |  @8 J! `7 \  Rkk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
1 @* a/ Q3 k. j( L+ C9 |* x! {  I> kk;
* }9 W2 n1 b! x4 s2 q- A/ }# bkk eq k;
3 u& Z' ]0 Y% r: t  e& v
5 f, E$ [. t' \* M2 L. bEltseq(kk) ;Eltseq(-1/14);
& Y! J* S8 U9 K! q- H; _+ |$ }
9 w7 |2 S; S# m( W% K

) e% U3 p. A% D
# G6 x; ~+ V8 A  z3 a6 o

: N' ~( |2 d- o5 z2 X" F+ z, i( ^
& _$ p, f3 {! e1 S2 Y. |) R! M
" p% K- x9 i+ u: x' X& W

=============

" w6 N6 P* u$ Q3 i5 ?3 T

5 ~# A+ [( I0 }
0 B* x3 w4 y5 r3 NInteger Ring
, v' o& S# M% r' g( x-1666666666234567890
-0x17213080A7E55CD2
' u+ v" }8 z$ C-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
, m' d$ R7 j" I-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
9 h+ h# s+ p/ N& a2 n4 b6 f2 i$ C5 o9 p0
+ x% k  u( a9 P8 D  v1
0
, A3 N# a+ R6 e7 H" |) z[ -1666666666234567890 ]
[ -1666666666234567890 ]
% y; j2 W: n- h3 {& W* V1
13
; w4 Q5 \/ P# [1
6 Z; `7 T  L8 _; [* j3 s1 k3 d
-1666666666234567890
+ ~5 ?) R! @% i! A4 N$ otrue
$ K& \  i5 @3 `( C8 S-1666666666234567890
) _+ J6 A# G/ H1 N1 G' ~9 Ftrue
9 o: z1 n: X2 o8 ?) F-1666666666234567890
; k- D4 K8 \# C/ r  Ztrue
, X" c( k; t6 i5 z-1666666666234567890
/ l5 U  A( z/ e7 ]# \2 c9 Itrue
" c- u& _' Q4 a& R1 N[ -1666666666234567890 ]
[ -1/14 ]

! D8 L0 l1 Q% G% o. b- @
0 g& Z3 T. v" t( x( G



+ }8 b+ H5 I. b: Y/ R: ~
% X4 P1 O9 Y, k% |2 _2 p# d  W-1666666666234567890
. a. b" y' g: g* ~7 @& B-1666666666234567890
true
1 k# o' n# [: [8 a& u8 U9 W-1666666666234567890
& P7 v5 b% M/ ptrue
6 ^: |- ^& D9 C' A6 I+ b-1666666666234567890
3 c! P5 l! C; m9 ~true
' g+ d; [, @9 \+ [" v, v-1666666666234567890
true
+ l7 X2 S( T* r. W[ -1666666666234567890 ]
[ -1/14 ]
' L: D; {8 v- z+ I) e  H

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
& V/ C" p- h; Gs:=0x12345678111;ss;
8 ^& o; w/ P9 q3 y+ J; x
sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
& {) Y. Y! E2 a8 ~4 g$ [Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
. d6 t, \4 b- j' \$ kSequenceToInteger(ssss, 2);
* E# R; z) S9 m* s: B6 |% y- ?. wssss:=Intseq(ss, 17);ssss;
  Y5 {, ~4 t5 y, f3 F0 O& e7 rSequenceToInteger(ssss, 17);
- C5 J0 Q8 |9 L! [* d7 assss1:=Intseq(s, 17);ssss1;
" R- E# O  P- \2 W9 F/ O: n7 gSequenceToInteger(ssss1, 17);转成2和17进制
3 a6 ]0 w7 b! ?0 Q
12345678111
12345678111
' M; @" m+ L; _5 I[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
5 K9 N8 b1 x# o' Y& ~& T: z3 ~12345678111
; q% `! q& J' \. [; g1250999894289
, A, @; b1 r' P' z0 K3 V1250999894289
+ D/ G3 H' ?5 c+ n[ 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
+ J: I$ C6 l8 b) R[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
3 o% S9 J  P/ y( g  D# r[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
" n/ j  `0 s1 x8 @% l! N1250999894289

lilianjie

Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;

; K! {) W+ x' S, f6 h% V5 t/ o0 }
# P- b& h7 o' x# O& wK:=Z!n1+Z!n2;K;

2 ~9 j9 t& M+ E( [7 N( k+ iIsField(Z);      是域吗Characteristic(Z);环特征
. Z1 h& M' S4 G) H  [: S6 PIsFinite(Z);有限环吗
IsCommutative(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。。。。
IsPID(Z) ;主理想整环吗

9 U1 J& Z. T- |* t, iIsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
5 f& @$ |. d4 L0 \9 yIsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
" X0 e, N' q( c( ^UnitGroup(Z);单位群
6 n) _5 d9 N" _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
  @- V: J: c1 ~0 ]. {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

& p4 S; U8 r$ I2 Z% j) }" \ZZ:=IntegerRing() ;ZZ;
  N0 F  e7 c/ ~6 ^7 gClassGroup(ZZ) ;
* n* G0 y# C+ I5 p0 @  A2 @
: c& a; M, N5 Q1 o===========
2 T" [% p, `, {
Residue class ring of integers modulo 5
0 k6 n% H1 F& }3 ~1666666666234567890
3 h/ u5 C# N; n1
9 }# l! ?- O! x1
; j# x: L6 }3 I% n3 Q2
9 w0 C3 t$ u: u9 d! Q$ e- q  ktrue
8 `3 E! f( A$ p1 ?! u. H5
2 Z' Z( d6 k$ Y; ]9 s* K2 ttrue 5
/ j3 i# h: T9 ytrue
false
) x& b- e0 e' D3 d( ?' Ytrue
true
true
; Y: D# s& c: H& i- w, Mtrue
true
true
true
2 a- Z6 D) G! Z8 l: t& NResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
% s+ H* o1 b; B5 |$ Y: hRelations:
0 Y/ [9 T$ G4 f    4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
RngIntRes
4 g, d5 h/ \* f" p3 \+ ~; UPower Structure of RngIntRes
, |( U* C, ]9 oResidue class ring of integers modulo 5
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
Defined on 1 generator
! T/ j% o- a1 s( P. H4 MRelations:
1 Q2 s: x% k4 F    5*$.1 = 0

>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

Integer Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
, M7 e5 f  h5 d7 Q UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
, g( n5 c$ O8 ~0 x% X
Z:=IntegerRing(13) ;Z;   
& G  e) U3 ^' F1 g/ e5 |1 L, v# H$ ` UnitGroup(Z);
2 {5 B9 H2 u9 j8 M  k2 p) JMultiplicativeGroup(Z);
0 P; F- z  `; E$ pCategory(Z) ;
7 z- W: j+ t, z4 p& N" {- U2 @PrimeRing(Z);
2 Y* F9 B/ v; B1 p. F  A8 a# CAdditiveGroup(Z) ;

4 l4 V; O- O7 \4 r
* f+ ?/ Q3 n4 `7 X) r
Residue class ring of integers modulo 12
& P& B- p( \! n5 _3 T- E* CAbelian Group isomorphic to Z/2 + Z/2
8 f" @$ \: |+ {& H" {) HDefined 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
( z  t. }6 l3 I$ b; ]' XRelations:    2*$.1 = 0
    2*$.2 = 0
RngIntRes
Residue class ring of integers modulo 12
/ X8 S1 u2 {& {Abelian Group isomorphic to Z/12
+ G- X, @" I% b/ u$ gDefined on 1 generator
# \  `) ]+ v+ l7 Z' S$ a4 FRelations:
# S: r! x; V6 I% Q6 u, ?! [    12*$.1 = 0
8 {( R9 t% ^# k. }- \: x% f" a5 tResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
Defined on 1 generator
/ n4 [$ F  k, t" u0 t( l8 D& x% [Relations:
2 ^9 r' j* {3 t    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
( ?: M; y- x4 l& {! LRngIntRes
8 r- A, s0 w! k8 c/ l3 O! rResidue class ring of integers modulo 13
  k$ Q; r; u" x5 d  Z3 P0 F! vAbelian Group isomorphic to Z/13
4 _9 }3 I0 }9 X1 v' A( uDefined on 1 generator
# h2 S6 T- U, h. Z, kRelations:
    13*$.1 = 0

lilianjie

! d9 g& n2 ]9 _7 ?( [/ G- ]9 `
- q9 o7 o0 M2 \- Z/ J* ]Z:=IntegerRing() ;Z;   
' [  k9 X4 }+ F0 {; o4 O
: w0 |4 F  _& }9 IR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   
" B( r' ^; G7 @6 i: X" z4 Z
! l! ^" z, ~0 X) f6 ]7 D' t
1 O2 S3 s9 N: B8 oPrimeRing(R) ;
6 b% @7 I( Z  m5 uCentre(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) ;
4 k% j1 l3 t9 h( s
IsPID(S) ;
IsDomain(S) ;
Has**(S) ;
0 V, }+ p# p9 P( T1 [R eq S ;
R ne S ;
2 H; r% U/ ?' |: d+ L/ C
) g2 @9 ]* \0 m* B2 f3 _9 eParent(R!123) arent(S!123) ;
! v. N, S6 c% ?' G5 U/ |4 |Category(R!234) ;Category(S!234) ;

7 [& b3 f9 Z( E  [2 Ja:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
Representative(S) ;

(R!a) in R ;
(S!b) notin S ;
' p" r+ P) A% W& v/ k  c1 ]$ XIsUnit(a) ;                是单位吗
0 w2 l; q7 y; l  XIsIdempotent(a) ;是幂等元吗
( p7 \  I# u- wIsNilpotent(b) ;是幂零元吗
; O$ R: _) w: W3 C6 U5 U2 @! pIsZeroDivisor(a) ;可除零吗
* c1 B1 P( @7 g0 r3 X5 OIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

" D; r6 f9 G2 L0 bZ!a gt Z!b ;
Z!a ge Z!b ;
Z!a lt Z!b ;
- m  x5 X; q8 b3 r* p0 ]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) ;
- s- Q2 I% N, b( z. Y9 ^
Maximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;



0 g5 f5 e/ j1 xInteger Ring
; v7 n( s" U" K+ Q- T6 }2 @6 nResidue class ring of integers modulo 12
Residue class ring of integers modulo 13
& N) X  P  l1 q% b5 TResidue class ring of integers modulo 12
- R, f# D$ @5 e9 x5 `- I, }Residue class ring of integers modulo 12
12
" D& ^; y: M6 u, _, w12
false
: r+ K2 x$ I8 |false
2 a. A" m; m1 k4 E' xtrue
6 `) O" r: n' y$ g, P; T6 W: T% s& htrue
. `7 a4 ]8 w- @" Etrue
true
6 }" }: C5 M( Z5 `, V$ y4 Zfalse
true
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
7 W6 H9 ^5 Q9 W% {: XRngIntResElt
RngIntResElt
9
12
0
: X( s+ L- L: r* F' i( e8 D0
' A4 N) U- H& M) ^  w% s! s6 E4 ktrue
% h& \* l# r0 \, z) i: Sfalse
false
2 i% P3 I$ z! j/ D) U0 \) y% j( r% Ltrue
, y- J9 U0 G+ j( z; w% cfalse
. l8 {3 i( R; O) x0 Y$ U( Qtrue
' h6 L  x2 X; y5 k0 _6 Cfalse
5 H1 B2 F3 c! E$ Rfalse
false
0 I/ H/ ]- m. [; K2 Bfalse
3 C9 e1 V: A  t8 C" etrue
( k$ m7 p% W- r+ g, s( ]( r- utrue
! Y' H* I7 k: ~12
' n' j# o7 G" |* H5 x1

>> Maximum(S) ;
          ^
7 k# C# o% b; j# ]! H; rRuntime error in 'Maximum': Bad argument types
Argument types given: RngIntRes
& F" @& K3 H4 c, G; K
- ]0 A+ s! |" k0 a# I9
3 [% s7 k% X+ t0 k) R8 Y
>> Minimum(R) ;
          ^
5 f5 M1 V6 J2 R" b5 X, vRuntime error in 'Minimum': Bad argument types
  {. V, I! H0 ]: XArgument types given: RngIntRes

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
9 s: V  ?% }$ Y0 gI12;
; I% H9 o8 D$ v0 W; vZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
IZZ15;
! V! v  X* P4 G1 f0 c+ A) T. }I12 eq IZZ15;
$ k3 N! q0 e1 a; u( ]Q1:=quo< Z | 12 >;Q1;
# v" j( f' b4 J2 a6 P( L6 N- YZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
IZZZ5;

I12 *  IZZ15;            理想和/积/并/交，
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，
6 C) w0 O8 X( d; q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，
1 f. J1 Q. r$ Z- N3 f. Y
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 g; C6 {: O( r# A  \' lI12 +  IZZ15;
, i3 K$ t4 Y/ g" ^/ ]) q+ V8 X7 DI12 meet  IZZ15;
( x" @- u" p9 Q# Y+ e: Q
! P7 H" u) X; n& W$ {/ X' m5 tI12 * IZZZ5;
. I9 O+ m6 j( c! ^# uI12 + IZZZ5;
! }2 \- H/ w3 \. |! r: qI12 meet IZZZ5;
I12 / IZZZ5;
0 H& K  R, g0 m2 {' x, _IZZZ5/ I12 ;
Z * IZZZ5;
% @& a6 c# t5 ^/ D* II12 + IZZZ5;
IZZ15 meet IZZZ5;
) f0 z6 l! |5 f/ Z3 J5 |) xIZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
4 j; u" R) w8 m( HIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;

I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
: Q% r2 S, \# [IZZ15 subset IZZZ5;
IZZZ5 subset IZZ15;
7 |. \5 K" ^8 ]- ]5 |0 \# C9 nInteger Ring
0 i# }3 p0 F7 u9 y+ ^' I1 LIdeal of Integer Ring generated by 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
false
Residue class ring of integers modulo 12
& }& |; v2 `8 v2 H+ i9 OResidue class ring of integers modulo 15
2 A& M7 _- |% S7 J3 B4 ]2 r/ z' M2 aIdeal of Integer Ring generated by 5
" k% a1 `% _) ^; rIdeal of Integer Ring generated by 180
" R" z. C0 Z5 r( IIdeal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
Integer Ring
Ideal of Integer Ring generated by 60

; k$ Y" }0 j& D, q$ e) ~9 n>> I12 / IZZZ5;
  W( [! v' t( f       ^
Runtime error in '/': Argument 2 must divide argument 1.


>> IZZZ5/ I12 ;
* A$ p) l7 `0 h        ^
( d5 Q5 y% i: M& TRuntime error in '/': Argument 2 must divide argument 1.
  O! S' V) v4 N2 V
Ideal of Integer Ring generated by 5
Integer Ring
Ideal 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
Ideal of Integer Ring generated by 60
# J( h# G/ b$ ?Ideal of Integer Ring generated by 15
! j1 o( R  f# E, `  V" wIdeal of Integer Ring generated by 3
Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z

* z' q3 C% S3 C- e* S$ J- e; E+ T2 ifalse
true
3 N( E$ L8 o& Ctrue
false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
5 ?9 F9 U7 [& Y3 [I12;
4 d; a  o& ~; X% N5 N& o0 EZZ:=IntegerRing(60) ;ZZ;   
: l# A# E& m% ]6 J) H8 O0 i. RIZZ15:=ideal< ZZ | 31 >;
9 i, d) G2 X. F6 V) ]; iIZZ15;
ResidueClassField(I12);
" T# l1 O7 h# j3 Q9 Q5 _. \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
! U) G1 m  Z* Y7 P+ r& R! mloc< Z | 19> ;
6 \& n; S+ x; Z* |+ h9 O( v3 Iloc< Z | 17> ;
. {! d+ O2 i/ P" W/ d! u2 V* ?! S6 ~/ iloc< 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 | > ;
& u) u. b9 R5 g& m7 Y8 [
ext< Z, 2 | > ;超越扩张到多元多项式
  D" }: G) @8 g) J3 Q
ext< Z, 3 | > Completion(Z, I12) ;

3 T: j2 ^6 @/ A! ^7 _
6 x- N  ]  ~, D. Z2 Z. t' rcomp<Z |I12  >;
2 z4 q0 \2 L- ]' b* r3 J    素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
4 p5 M- I! l* Bcomp<Z |0  >;
" s) i  h, ?" Z$ ]
/ j' j2 O5 R8 jInteger Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
' ]( T/ y5 v+ S" [$ D9 E7 qResidue class ring of integers modulo 60
$ T; U' p/ V) jFinite field of size 13
! x! y2 `! ^: z2 X' x1 L) G" B/ xMapping from: RngInt: Z to GF(13)
3 y+ D7 b$ k9 ]/ @. T6 A5 bmodulo 13

/ ^. D+ q: l5 s& ]' U>> ResidueClassField(IZZ15);
2 ]1 n! J; R8 F# ?! O                    ^
: o+ B3 q, ~8 t/ Q( p6 LRuntime error in 'ResidueClassField': Bad argument types
: W3 |+ Z4 d" [  ]/ K  [' mArgument types given: RngIntRes

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
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
* p% }! @" N7 EValuation 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
Univariate Polynomial Ring over IntegerRing(60)
6 @& P! B' {0 }% e  ^( D, p
>> ext< Z, 2 | > ;
4 C% l: X2 j( |) ~      ^
! Z8 g9 p5 a) {0 YRuntime error: This constructer is no longer supported


6 d( E! |+ `& N" L2 m0 Q, A# E7 C, `>> ext< Z, 3 | >
2 A( i  t* i! E3 y" K      ^
5 x) _3 j2 U. }% u) Y* n/ f; gRuntime error: This constructer is no longer supported
: ~5 x9 D' \) v$ @
13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)

Completion(
; }4 g  v' W9 X, s: b' _8 D    Z: Integer Ring,
$ N7 s  G; e3 L    P: Ideal of Integer Ring generated by 0