一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
9 s! ^4 a6 }9 b. W> n;
" s/ H# }# W3 J$ j; W
9 g: Y  W' |. ^% B9 i1 |' B& g6 V> n:Hex;                           转16
IntegerToString(n, 2);        转2
/ ]( X% L: ~, U) tIntegerToString(n, 10);       转10
0 B! Z5 d% B8 c5 J0 v" WIntegerToString(n, 16);        转16
0 q3 Q& A  ?3 ~IntegerToString(n, 36);         转36IntegerToString(n) ;
  W& i- w7 n7 _3 RIntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
Representative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
4 [; U9 E' H" G* }! L$ S! @
7 P6 A0 F% b; c7 S/ h% lm := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
5 ~( ^# p' g: N  ^/ B8 O/ y# p> k;
) S; c7 O9 s" o3 o! y. D2 d4 sn eq k;
0 @& P' k, |% L- Dkk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
$ T2 H5 d/ G* f- _kk eq k;
5 W$ v" F( q! \) g8 c6 L
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
! O" C4 n  |3 u' w> k;
1 F! W( P! H5 b) n* R- b/ an eq k;
! \/ t! T6 _; G( E/ hkk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

, O; n  r; _7 d' }* [; D- `8 uEltseq(kk) ;Eltseq(-1/14);

4 [1 q: v/ N1 c7 [! f7 k

; w: J# }$ m* l
% {" B' n/ R* f% A3 v, ?1 H. D5 xk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
2 I& ?: S. k0 C3 H) X, `> kk;
/ k$ Y6 g# }! |" O! t$ gkk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
0 ^0 u! a( V, u+ u, H8 ^n eq k;
$ N7 L) J; \2 S2 y. M: Bkk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
- U! Q4 {) V; g6 Z
& r9 X3 V) r& E1 N9 [/ q' vEltseq(kk) ;Eltseq(-1/14);

' ]  N4 N' U8 {: e5 D6 H& R

% R! d" d* S" C, S$ `  Y' c

: j' @4 [/ U  L3 m- [9 T5 r
% }9 Q; n  u+ P2 q% j  `


3 r" q' t4 R. i; d; ^! u+ N
% p" g* k, r) ^( e=============

; R2 S; z! p( H


/ n5 U$ p  f+ g  A. h3 \! f5 T9 MInteger Ring
-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
% p' }" x2 w; Q+ D# c& ^/ M  V-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
! U4 \9 s! p* C. ~-1666666666234567890
% u* G6 o' U* u- X0
; M8 _5 F4 v. s, d! Y0 S; P1
0
[ -1666666666234567890 ]
[ -1666666666234567890 ]
7 ^1 F( D4 B( T& i+ A; {5 d; u; c1
13
1
5 Y* x0 A0 u  s* {6 m2 N( c/ `2 B
-1666666666234567890
9 t8 l6 B1 A  R- C# `/ |true
& E7 s* Z" W6 _. x; a/ Y-1666666666234567890
! G# S1 |# }+ Y# n+ P2 {& d% c  Rtrue
-1666666666234567890
true
1 R9 p  j+ u7 ^3 Q-1666666666234567890
4 N2 S/ t" j; B6 K$ N: ~true
[ -1666666666234567890 ]
8 p( d1 X/ H3 _4 P/ v# ^4 R, N[ -1/14 ]

# ~) P& p# {& s' x
; t/ M+ s& X, h+ J4 b& `$ ~

$ N# s8 c  H  z. t* P' n
0 f# H4 r3 T" {  v  D& J

-1666666666234567890
! p8 f6 ~( y* J1 A1 F5 S% @-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
true
6 V" P7 H& M+ G% O6 I  G+ _- C-1666666666234567890
true
! k- R' }9 [$ p/ V5 p2 L. A2 r[ -1666666666234567890 ]
! v+ g+ l9 v/ A, ^9 P: @& g[ -1/14 ]
( I9 F- x4 `8 r* Z9 C# E( o

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
+ f. A4 h* t2 B, X. S: M* Z, Ws:=0x12345678111;ss;
1 I) S% [: C/ D& k% O2 X/ L
1 S3 n2 T: G  Z) R8 B' F& Tsss:=Factorization(ss);sss;
  f( x- C8 \# q4 D& d! E$ \sss1:=Factorisation(s);sss1;
2 J$ U/ G1 [$ Y+ \FactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
2 L. H1 q: g# t8 yssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
. T$ @3 p: E* A0 q. u  kssss1:=Intseq(s, 17);ssss1;
- d6 k$ k+ R+ r0 I1 V+ mSequenceToInteger(ssss1, 17);转成2和17进制

12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
, y' K& A4 p' p/ |4 [6 L# S[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
% d& j! M; D5 P1 O! s$ \12345678111
) l( a, r5 d% I1 t1250999894289
% c  q/ F4 e+ a, B% O1 {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,
- q+ f! d5 ]- @$ Y" \1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
% O6 t  v- d% z3 G" l) S: E3 U: a12345678111
( D. a, ^! t! q[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

lilianjie

2 U* B( w. K9 c' a- b
6 T. ?  G1 ^1 jZ:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
n1:=Z!1111111111111111111111;n1;
8 @6 k$ x* @/ m5 S) \n2:=Z!11333331111111111111111;n2;

- \9 ~) Q& \' h# Z- n
K:=Z!n1+Z!n2;K;
; W7 J/ D/ Q- ^& V
2 j9 \" ?: ]0 V7 X) pIsField(Z);      是域吗Characteristic(Z);环特征
2 J7 {* D( @, ~- ^IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
6 N2 m+ B/ s6 w) C6 D9 o4 JIsOrdered(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。。。。
# \2 T9 J. U* Y5 [4 dIsPID(Z) ;主理想整环吗

IsUFD(Z) ;唯一分解吗
( [1 N  g% }+ N* h7 n2 eIsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
0 x* M5 I7 ^7 d: N& [IsPrincipalIdealRing(Z) ;主理想整环吗
IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
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
+ F$ k' a. _; ~+ [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

ZZ:=IntegerRing() ;ZZ;
ClassGroup(ZZ) ;

===========
5 c) j: I4 S; W1 [, s: ^
$ v" i% V9 d" }6 k! J! [8 \% lResidue class ring of integers modulo 5
1666666666234567890
* d) X2 A, E4 y6 O& Q( h( i5 s* y1
" a8 ~( E' U/ z5 V5 l" \- o. Z- H1 `8 l1
7 S. i( ^; g: E* y2
! o3 g( q# @* j, R0 Xtrue
5
, @& u4 O  a9 Dtrue 5
6 ^" _/ P$ ^9 [* qtrue
false
, n2 _  W3 c& F1 }  \8 A' Ftrue
2 T) B0 Y$ b2 O% i7 x, f6 o" `true
true
: N1 @* e2 [, {) n( G4 G7 [true
true
true
0 E7 X6 S. J% R5 V( Rtrue
Residue class ring of integers modulo 5
# |# O) V, Z! j- S' _! q+ ]7 p, HAbelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
) Y1 d) M# M: t5 n3 Q2 @! K    4*$.1 = 0
7 R/ a: R" U& TAbelian Group isomorphic to Z/4
+ W% c: U' m' r8 R5 hDefined on 1 generator
- z. D1 v* ^. ^+ b0 A: E! ^& L- k' cRelations:
) Y- X! a! p& q( p4 X8 _    4*$.1 = 0
RngIntRes
6 N( x/ S9 v. A2 U4 vPower Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
; Q) a2 s3 T( g0 b- o& v+ T1 Y# u" `Abelian Group isomorphic to Z/5
3 `  V  x. o  _2 Q; j2 JDefined on 1 generator
' |% X7 l# p6 Q( URelations:
4 s9 Z& n' ~0 M5 K% ^    5*$.1 = 0

" ~* X! @& F2 x" |$ _>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes
7 B! S: F- C* t7 b1 U
Integer Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
MultiplicativeGroup(Z);
1 V1 v4 e5 ?# B0 S- z3 m2 @Category(Z) ;
% s7 f4 n# C3 a" D) N/ m; IPrimeRing(Z);
AdditiveGroup(Z) ;
4 C" U6 s2 J& w9 y0 _
Z:=IntegerRing(13) ;Z;   
6 d1 K; P4 A& N5 L0 Y" \: P- u UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
& g9 q/ W# |+ s3 {7 p0 ~# _1 R: zAdditiveGroup(Z) ;
* _! `8 f. S5 N


0 W3 c- X& r7 a5 [; p4 n( `! Q, EResidue class ring of integers modulo 12
/ ]0 {) k2 G* TAbelian Group isomorphic to Z/2 + Z/2
3 s1 y8 g% _0 x5 m0 X9 @Defined on 2 generators
Relations:
% Q3 g( F" B, A3 I1 }- X; d3 M3 [    2*$.1 = 0
    2*$.2 = 0
6 |& g7 z! X# w2 oAbelian 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
Relations:    2*$.1 = 0
) L2 c  w8 c* R. S    2*$.2 = 0
% m* g- \# r4 P  {( @, HRngIntRes
Residue class ring of integers modulo 12
9 y6 o, F5 h' S5 i) Z' ~/ LAbelian Group isomorphic to Z/12
" T- [; V4 x7 u3 s8 ZDefined on 1 generator
Relations:
    12*$.1 = 0
$ C5 i! {( O9 @  }+ B# T: |3 G6 A( _Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
( \- S  b- c5 n3 X4 V    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
8 q9 Z) r+ p. v6 k/ k    12*$.1 = 0
2 a+ ~4 A2 b9 Z0 hRngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
( T9 x2 T, I. e1 Y7 `2 DDefined on 1 generator
Relations:
    13*$.1 = 0

lilianjie

; I, e; o6 t( i& @3 j
6 z! Q! C, `; {1 @Z:=IntegerRing() ;Z;   

R:=IntegerRing(12) ;R;   
* r- W% u/ t1 u4 oS:=IntegerRing(13) ;S;   
$ q" p% R) r3 W: `( S

: ?; ^3 h& W6 _PrimeRing(R) ;
Centre(R) ;
" A  m6 G8 N& Q& Q$ T" t) A$ a
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) ;
3 a6 _' M$ U) E; k" bHas**(R) ;

IsPID(S) ;
8 X5 ]8 g  b; \5 [" T; r$ @IsDomain(S) ;
0 v7 d5 b5 r8 hHas**(S) ;
R eq S ;
( t) `$ ~( ?1 f" y+ NR ne S ;

Parent(R!123) arent(S!123) ;
Category(R!234) ;Category(S!234) ;
( p( r' Q6 b) V3 I- t* Z, p) H
! Y4 ~. N4 N1 j5 C) x: T+ X# q+ Ia:=Random(R) ;a;b:=Random(S) ;b;
0 p& [& |0 E& G% H( SRepresentative(R) ;
Representative(S) ;

' O( ~) A6 H: U# W; ?(R!a) in R ;
(S!b) notin S ;
& m% t$ ^" |7 A" ^IsUnit(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
' w. Y8 F0 _. ZIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
. B1 b7 A& J+ [! T+ b7 t
, W# T! I% z4 e0 z' c6 zZ!a gt Z!b ;
. E& j6 e) K/ zZ!a ge Z!b ;
  h; h; l2 q/ KZ!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) ;
1 i; l- s" y- y: d! ?% qMinimum(Z) ;

% N" w$ K1 Z. e5 d$ e% ^Maximum(S) ;
Minimum(Z!a, Z!b) ;
2 m3 Y8 t- f8 x' zMinimum(R) ;
/ d( J3 W  j; E  T% V


Integer Ring
, w6 G" k. E5 k0 ^7 O1 MResidue class ring of integers modulo 12
9 F/ ]; T6 t. W/ j+ MResidue class ring of integers modulo 13
6 h7 t; ]' J- ]3 F6 @3 MResidue class ring of integers modulo 12
: n) h$ j: M, D! W8 jResidue class ring of integers modulo 12
0 k) T7 S8 U8 `12
% ~1 O  S# @' \" ?12
1 f* a% p  D% F, jfalse
false
7 H+ t* y8 [! V/ |2 ktrue
true
/ [* ?5 N/ m" G& i& e1 h9 Dtrue
( ^1 c  w' \: ]( d0 mtrue
. X( V: |+ x3 j. Q9 ufalse
true
: ^' j  j* U% n/ Z7 @4 z+ b+ \/ J# ~Residue class ring of integers modulo 12
1 R2 w! P" Q1 s# [Residue class ring of integers modulo 13
RngIntResElt
2 E8 a- y& D" hRngIntResElt
9
12
- |0 V* `$ N4 q1 c! ]/ u; \% E  c0
0
true
false
false
7 d, |' [1 l9 ?& I6 r' Etrue
false
- F6 A2 D) C7 j* f5 S2 wtrue
( q1 D& p! ?" n7 L8 i9 Y, ^" dfalse
false
/ R9 K: \, Z+ P! D5 i3 X. Yfalse
0 h; S: b' F8 R: R/ Ifalse
8 ~$ q& K( s& S% d( s: O) Otrue
+ ~: O7 h( U% v* r  M6 y. Ktrue
12
1

: T" u( I, a9 [2 A$ U0 {$ {! D7 D>> Maximum(S) ;
) e2 `4 @! \- v# T  F& m6 [          ^
Runtime error in 'Maximum': Bad argument types
( f3 {. }$ C4 J3 {# W5 i$ DArgument types given: RngIntRes
8 T6 ?0 w# U1 h, q" Z# h6 N' [# w
9
1 N7 G8 J1 U+ @
7 {( V5 R% ^& P2 B) }* E/ C>> Minimum(R) ;
          ^
' h, T$ _9 x% E5 KRuntime error in 'Minimum': Bad argument types
# k. L: t2 y. EArgument types given: RngIntRes

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
. \7 W7 M; E: d$ k" n% B# LZZ:=IntegerRing(15) ;ZZ;   
0 \2 z1 |* k% q, ^; D4 b, hIZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
( B! `8 z2 b; A  X) U; o: WZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
/ k0 K; F5 X6 d$ Z( S# D; q; IIZZZ5;
, V% O5 z4 T& A
I12 *  IZZ15;            理想和/积/并/交，
+ ]) [! X+ [7 d+ O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，
$ V$ b5 c8 f% u* J7 g, R理想并就两（可多个）理想元素并，就不一定还是理想，
: K3 H: _, I) N5 i理想交是理想（可多个）元素交，理想交一定还是理想，

* O( [$ k0 I) Y$ l! g9 [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
$ d+ B6 C# q1 r6 R4 D3 i0 KI12 +  IZZ15;
I12 meet  IZZ15;

2 X  M9 s& s5 Y6 ?& [9 x$ RI12 * IZZZ5;
I12 + IZZZ5;
6 G$ C* H: G; y: GI12 meet IZZZ5;
I12 / IZZZ5;
3 [6 B: ^6 o" r  p. xIZZZ5/ I12 ;
7 e& r+ a# }. _; f& kZ * IZZZ5;
2 E$ [; x% E( p0 \: m$ JI12 + IZZZ5;
IZZ15 meet IZZZ5;
# q1 j$ c5 u; v) j) ?" cIZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;

I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
2 U' ~/ u/ W7 l9 XIZZ15 subset IZZZ5;
& G' l- `7 t7 O. pIZZZ5 subset IZZ15;
# K4 a% [9 n+ m& \Integer Ring
+ G( a$ S4 m% C5 \4 gIdeal of Integer Ring generated by 12
Residue class ring of integers modulo 15
. u: N* k# S% U8 F, b9 BIdeal of Integer Ring generated by 15
; x  o  g6 g/ g" ?! c& Ifalse
: M5 F- n  P" ^/ }8 i% vResidue class ring of integers modulo 12
Residue class ring of integers modulo 15
+ J+ {% {- C% P" U- ?Ideal 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
/ f* h& y2 O0 s- y6 f+ [8 d( yIdeal of Integer Ring generated by 60
; t  w, X( A& S* z, B) y3 nInteger Ring
) @$ P6 u1 n+ o/ [( `' sIdeal of Integer Ring generated by 60
5 z! U) G, h; [- ~
5 Y- F: w. X* Y$ ~& T>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.

% d1 u: Z( g% Q) M" @# y
>> IZZZ5/ I12 ;
        ^
- _7 \. ?/ n+ w: v7 w0 R6 [Runtime error in '/': Argument 2 must divide argument 1.
5 f; k- L3 R# A  P
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
3 b4 _+ K- U0 `- E2 n% aIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
0 B$ J5 D( @, aIdeal of Integer Ring generated by 3
Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
- C4 u1 R" c" `9 |! F7 h
" z3 h- D  Q. z% \- ufalse
& G3 l/ Y: p# E3 Ntrue
( n8 @7 e8 i1 S9 {: h. J. atrue
2 v! e) W% A% x% d; M7 B7 ]false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
I12;
( C- w: T2 k/ G6 Y$ a- y( D6 Z: XZZ:=IntegerRing(60) ;ZZ;   
' K7 }1 `; z. ~* E* _. KIZZ15:=ideal< ZZ | 31 >;
IZZ15;
ResidueClassField(I12);
- B! M9 D! R$ X: F" I1 i# A) }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
. r# ]8 ^' x- [$ _! X" V# H, b$ lloc< Z | 19> ;
3 O. n3 I1 ?+ l/ aloc< 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 | > ;
1 T1 z- O: L# E+ b& x: R
( S- d5 d* m; Aext< Z, 2 | > ;超越扩张到多元多项式
0 `3 L$ _9 p5 N
( H8 G  v( [7 T& next< Z, 3 | > Completion(Z, I12) ;
0 Z/ l8 g8 r& P0 z/ B5 r, B6 n

1 |5 A9 f) ~% \6 Kcomp<Z |I12  >;
3 C) _' c! @$ K1 i" G    素理想零理想完备化，和P进环联系起来
  t' @$ ]1 }: K: @* eCompletion(Z, 0) ;
comp<Z |0  >;
1 R7 ~9 {. }3 P* A8 i
Integer Ring
, t" ]8 l. V$ S/ c) K! H% F0 PIdeal of Integer Ring generated by 13
5 f8 H9 t4 [4 y3 Z6 m( g2 a0 DResidue class ring of integers modulo 60
8 l% [; C. Y7 ?4 QResidue class ring of integers modulo 60
- R. |/ ?9 Y! j* h0 v8 UFinite field of size 13
: _. _2 g  X# F6 bMapping from: RngInt: Z to GF(13)
7 D* N: J7 P2 Vmodulo 13

>> ResidueClassField(IZZ15);
                    ^
Runtime error in 'ResidueClassField': Bad argument types
" @: h/ d" S  o; tArgument types given: RngIntRes
% F# A) O3 O# m
2 t# ^$ x( t( k! LValuation ring of Rational Field with generator 19
) g  h  \$ ]- kMapping 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
$ x9 y# i& S5 ^' I) d% D+ G% MValuation ring of Rational Field with generator 131
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
4 n1 f6 F' E. o$ bUnivariate Polynomial Ring over Integer Ring
0 @  K, y2 ^. P: N$ }8 H1 X0 V( nUnivariate Polynomial Ring over IntegerRing(60)

>> ext< Z, 2 | > ;
2 }6 a6 {! N, T+ e% v# ~! \      ^
* q* x* C+ c% ^, _2 `( w6 d/ s$ eRuntime error: This constructer is no longer supported

. G5 h1 G$ T$ x, g1 V& L' t7 D
' j% Z6 w. @/ i) ]; w3 i>> ext< Z, 3 | >
      ^
Runtime error: This constructer is no longer supported
8 E' T' H* r9 T. f8 M
13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)

Completion(
    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0
- i5 v" B7 b5 e" N: g