一些初等函数：

lilianjie

:=IntegerRing() ;Z;
) L9 a3 ]8 r; @4 a' J. S( @7 x/ in := -1666666666234567890;
> n;
; p& A$ m- l* C# A( O4 C3 u" R
) V% ~/ {# X0 B  h( Q> n:Hex;                           转16
  K0 _" _7 A7 R2 L: P1 a( {IntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
! @( r0 R0 \$ p) M& G9 }  \$ g5 e, dIntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
5 T' p; u) ^+ n2 ]3 @/ ?Representative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
" U+ ?5 N( @. s$ i( `* i1 J. w& l6 c, f
m := 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;
( @2 M; z7 T5 n! _6 `n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
* z% I( ?$ Y. U9 g4 m  s& E> kk;
kk eq k;
) D# h1 j1 X/ Z  k+ T% K( F4 M) o
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

Eltseq(kk) ;Eltseq(-1/14);
" m# C: Z( J3 z, L
; X( o/ y6 L1 ?# }2 p( d: x4 G9 f


/ G+ T7 c" N: H# `/ ~% Vk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
3 O8 l+ d6 y1 g. n8 W/ n, q> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
5 l. h/ Q' _2 F3 ykk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
4 L5 ]* Q5 m$ i; kn eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
5 R3 Z# O4 |1 v/ k+ S7 x- u2 l" o> kk;
2 T. p. Q5 r2 Tkk eq k;

9 F: Q2 u  P  R1 OEltseq(kk) ;Eltseq(-1/14);


  R) O; j2 c4 a: `5 y  H6 f


' [6 A' d! ?: _8 r2 }0 D
  F* i# ]9 F8 L: p9 C+ T% `+ y

% V$ E! c& j& T! D1 b# a. A* |' x8 `
9 r1 z) G8 M% J* ^6 k
=============
9 R( C5 w% F- Q4 A- D% x7 i% i* X. s
* E& M2 N; V: D9 W4 f$ w; y

3 U; _( g. r) u! n
Integer Ring
) s3 c' W+ M8 V6 u" {- K& V-1666666666234567890
2 |7 E$ w2 |4 w7 l-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
) ?& v: i( G) N5 u6 m' b-1666666666234567890
-1666666666234567890
0
2 D7 a: e5 `& w- r% |) c$ i4 z1
/ F4 D/ e4 U$ x/ @! J  o6 N0
' w& k$ X+ W  q2 y; E[ -1666666666234567890 ]
9 z1 B* }3 _5 f7 F[ -1666666666234567890 ]
$ ^5 p' _8 H* @, \# q1
13
: U: j: ~+ g5 Y7 P1

8 h9 Y: V) N+ G+ e- ~-1666666666234567890
true
-1666666666234567890
" O9 O& @6 V1 G" u! {! Ltrue
-1666666666234567890
true
-1666666666234567890
true
6 C5 q8 G, z- N[ -1666666666234567890 ]
' c  l: o& N1 }7 W5 p, t1 c[ -1/14 ]
: v' f% D  x; f' ^
- h$ ^# U' S. Z, f/ x+ ^
( ]! e' Z+ r4 b* Q$ j
* B9 T) ]! ?" A( [' q/ f0 t
/ C# V9 ]: K2 p/ @7 }
; W* p( n$ }+ O- A1 z  K) x
* z( p# E4 X# j! _- }
$ x7 e" a$ r$ v0 A9 G# o-1666666666234567890
-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
1 F! U8 Z% w% jtrue
# ~* A" ]! P3 O1 Y" Z-1666666666234567890
true
% q8 W/ A- \; q8 [[ -1666666666234567890 ]
[ -1/14 ]

( E3 [  a- v8 d" q

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
4 e7 K' H( Z* b* S8 M9 |s:=0x12345678111;ss;

sss:=Factorization(ss);sss;
) T8 A2 T( K( C. l2 u; z/ [sss1:=Factorisation(s);sss1;
) f$ H; l5 ~6 Q; m: yFactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
  w* X: l: Z. cFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
4 X3 l0 K# t( ]0 }ssss:=Intseq(ss, 17);ssss;
$ y9 q! \) p# O* c: J7 K, J! g2 ZSequenceToInteger(ssss, 17);
8 _$ c/ m1 A1 y2 Cssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制
6 M6 t% Y' R/ M- O& j
12345678111
12345678111
/ x7 V, J* H4 [/ m. g7 Y, h4 {[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
' S$ [; [6 ?5 E0 B' ?  j[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
9 w* m! l, [1 a; G3 b- Y12345678111
1250999894289
1 D; M- F  R# p, L4 w1250999894289
[ 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 ]
! s, E! c8 S/ h2 n* l12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
3 e3 {8 S& v' \; N3 f; z[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
: H7 l3 @% B5 V- W1250999894289

lilianjie

, ^  `* }: B- \) ?Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
, v9 N; U% a# f, H5 m> n;
2 x* u9 P& N0 M! y8 {. Q! \) S! ?n1:=Z!1111111111111111111111;n1;
% ?7 b' Q4 `9 Y1 xn2:=Z!11333331111111111111111;n2;

/ X# o: H! N9 j( k) P
K:=Z!n1+Z!n2;K;

" P4 Y( Y& Z! I0 mIsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
# |7 r7 p+ S9 JIsCommutative(Z);可换吗
/ i' s0 W6 J+ A" ^1 i: W5 `9 L  {' m4 Z3 dIsOrdered(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
0 q: h( B& W1 h/ R+ O
5 n5 b" r6 f! \" @IsUFD(Z) ;唯一分解吗
* X* D2 l3 X4 W: s7 P$ g0 qIsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
: _3 r- a1 F0 i- p6 Z' |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
4 a0 W2 m0 e. x) S- S+ G) 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
1 r* |; z) G3 \+ o5 N$ b% e7 e
ZZ:=IntegerRing() ;ZZ;
1 F$ d6 k3 C; E1 ^" {ClassGroup(ZZ) ;
& n; u  Z; h. I+ M0 w( {8 A1 k
8 X$ p* o* m8 v/ e2 C# |===========
2 L1 O. f5 a6 S( ^: K
Residue class ring of integers modulo 5
- h7 c  V5 i5 Q8 Y  A% [! \' W1666666666234567890
1
3 `8 `/ ^- f6 g+ i1
& u+ b; o/ ^0 S9 q9 Q7 E4 U6 H2
$ d7 h* U7 Q$ f  n3 t  q3 ytrue
5
true 5
2 P  X3 W$ w" M1 Atrue
* _  I# z: L( C! @false
; f: T2 t$ c7 E# t$ P( |true
' I; _; M; d9 Z, t" ctrue
true
true
# ^5 w# L6 F2 Ttrue
; {) K! {% j/ C" K5 }3 }' Jtrue
true
8 c7 c( K- O* a+ s" ?Residue class ring of integers modulo 5
  [8 S. e, F0 k, x: ?+ i- \/ UAbelian Group isomorphic to Z/4
- a) R! i. I& m0 gDefined on 1 generator
Relations:
    4*$.1 = 0
8 e" @! L' n( t: I2 {: wAbelian Group isomorphic to Z/4
# ^; K1 w0 ^; V  N: `Defined on 1 generator
Relations:
    4*$.1 = 0
RngIntRes
Power Structure of RngIntRes
0 Q6 h6 n$ y! nResidue class ring of integers modulo 5
% z  x7 K1 ]# [4 I+ j; s0 T" NResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
+ @5 e" l' I# @" n, FDefined on 1 generator
5 }1 u6 s! D0 [# J) NRelations:
    5*$.1 = 0
0 X# T- [$ g, P7 Y4 k2 S+ P5 e( H
>> ClassGroup(Z) ;
+ V+ b8 A5 h( h0 {3 b$ V             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

6 D1 z+ F1 `) ~! n7 R: h; jInteger Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
  H/ g6 V9 K1 R! M* z3 ?8 x7 UAdditiveGroup(Z) ;

/ j8 r$ u. \! }) _8 y- X, r  @Z:=IntegerRing(13) ;Z;   
" [2 X* y3 C: O UnitGroup(Z);
MultiplicativeGroup(Z);
' V' ~8 W' Z" \, y/ J6 u: u5 U7 TCategory(Z) ;
2 @) t; G1 w, P; L5 m8 f4 SPrimeRing(Z);
AdditiveGroup(Z) ;

$ C; X/ V; O$ p2 o  t7 X
9 f- A4 a( s- K
3 [+ A: O( }! F# K* HResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
# D7 Z3 }' M0 V    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
Relations:    2*$.1 = 0
% M, ]9 v5 V4 G/ ~    2*$.2 = 0
RngIntRes
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
0 N' M) Q4 R6 }- D+ r4 [( qDefined on 1 generator
, ?  X) U9 Y1 ORelations:
    12*$.1 = 0
Residue class ring of integers modulo 13
3 S, o" X$ a! U: eAbelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
* y6 V3 z& L4 n$ r    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
- M5 X- k6 X5 P0 q    12*$.1 = 0
8 x4 l, k! ~3 s  ^! x: HRngIntRes
Residue class ring of integers modulo 13
# G% D& Y9 ]6 qAbelian Group isomorphic to Z/13
3 F/ h  Z6 ]1 |Defined on 1 generator
* F) ?3 e& z' r/ ^; mRelations:
  b1 N0 N1 ?- C6 H8 i" R    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   

$ b7 _. c0 q) ^8 G0 V3 Z' qR:=IntegerRing(12) ;R;   
2 O  u/ R8 s4 m2 @) yS:=IntegerRing(13) ;S;   
' d( O& Y4 k6 t7 t: e+ Q
: _& B& }9 o3 v3 j' v/ [
PrimeRing(R) ;
" |8 D) `; y+ s: C, DCentre(R) ;

* @) l1 r1 h! }+ M$ F  x3 kCharacteristic(R) ;
! C. `( N7 r. R4 ?* q5 }$ s" W# R ;阶----元素数
: B: ~2 l) f7 d, W0 {  @! vIsPID(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) ;
1 H. V2 Z3 a8 _& o
. Y6 X3 u" D! Z; x; u2 ^IsPID(S) ;
IsDomain(S) ;
Has**(S) ;
R eq S ;
R ne S ;
& v7 P0 w! Y7 D6 z+ S" U9 q
# `" S  M5 L2 m# L+ x1 XParent(R!123) arent(S!123) ;
, s2 r5 n$ a. o# w. a+ n& ^Category(R!234) ;Category(S!234) ;
% W; L3 {1 n% S! V$ M1 b  h) \/ X
: }9 h% N2 b+ ?' K2 oa:=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
$ F! \& A' d* o4 e8 s& F0 \IsNilpotent(b) ;是幂零元吗
0 U$ E+ E7 R2 q& P3 I0 _IsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
6 @, p8 h# z# f! Y' J
- q; L: {) b/ ^. [' nZ!a gt Z!b ;
Z!a ge Z!b ;
# T( X& j/ t- w) T: ]6 {) ~3 ]Z!a lt Z!b ;
- V" M+ h+ D4 c! SZ!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
$ r$ A7 X( z& {; S# V. Y  xMinimum(Z) ;
" _3 t$ P* C+ V& `
Maximum(S) ;
1 o# \# f3 F( OMinimum(Z!a, Z!b) ;
* r2 \1 j$ [4 {8 Y; R) i/ Q5 CMinimum(R) ;

+ X3 e* `$ _; J+ D1 {8 S$ v% ?
2 M" h$ c7 a5 @7 G; D; F$ K
Integer Ring
% p1 a0 C$ A- E& ]0 LResidue class ring of integers modulo 12
7 O9 r- n1 W9 q* nResidue class ring of integers modulo 13
' ~) D4 z! }5 O7 QResidue class ring of integers modulo 12
9 L$ y$ E) Q2 L" ~Residue class ring of integers modulo 12
12
12
& m) ]* }; m' V  T2 u$ \false
false
2 ?2 e6 t6 R& Gtrue
; k9 P0 T" e5 _) B- _true
true
true
false
true
Residue class ring of integers modulo 12
- z! H4 ~) I( `- B( ^Residue class ring of integers modulo 13
* C+ v4 C  B" _4 e, x6 aRngIntResElt
/ p" N  r* j7 L& I3 P$ XRngIntResElt
5 z" \& M; t) n/ e! o# `& F7 e9
12
7 `  X: x' F2 j# j; t0
0
true
# ?0 s# J3 M7 M5 _& Qfalse
. n& i" E, h% ]2 W, c# Sfalse
true
false
true
, \7 X7 E$ o6 V! \false
false
# z% n3 R$ F3 l' X; K8 pfalse
false
$ e( M; y/ s5 w- z; C* \. m2 k* R2 Gtrue
; Y' V7 ?: m8 I- o! wtrue
12
1

8 S. a5 i8 c5 v* }7 r: B' B) R8 l4 F>> Maximum(S) ;
/ N: O, ^- k3 R! f          ^
8 |  }/ W0 ^& I$ ARuntime error in 'Maximum': Bad argument types
Argument types given: RngIntRes

. l. o0 ]/ a1 i9

$ W6 E& E2 v) b>> Minimum(R) ;
9 T; u9 k! U) Z: E% R( y0 b          ^
Runtime error in 'Minimum': Bad argument types
! O0 R; ]: b0 p% w( \0 w! q' AArgument types given: RngIntRes

lilianjie1

, ^! t% q' |6 T- u! MZ:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
$ c0 m. R+ L( d2 P8 g! II12;
$ R' P& o& k0 O( DZZ:=IntegerRing(15) ;ZZ;   
& Z. \' G9 P" B7 w: L! s: ~IZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
6 O7 {. E2 H& }) W$ X7 E6 X9 r) g# RZZZ:=IntegerRing(5) ;ZZ;   
% C, h2 c( k! Q# QIZZZ5:=ideal< Z | 5 >;
) t7 x& [( S! _% }: c; L$ kIZZZ5;
5 K1 k4 k: `1 f+ N8 R/ m1 q6 Z7 b' \
I12 *  IZZ15;            理想和/积/并/交，
& f; Y  N+ L4 f  V$ p3 r# t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，
- D, V) @6 i% G3 p# F  o( y3 b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，
* r0 c' X- c+ h! R) E7 X  E  ]
+ G. ^  V: [$ U2 L理想积是理想交的真子集，极大理想交是理想------J根
! h" i: g& j! f' @" ?3 Y理想商就理想间同态：是必须能整除
( p/ B. Z% b* q' F8 y; h9 KI12 +  IZZ15;
I12 meet  IZZ15;

7 c9 S8 ^2 x1 T: t  gI12 * IZZZ5;
( I8 x4 {/ v" e! SI12 + IZZZ5;
3 e7 d- ~3 ]: q! sI12 meet IZZZ5;
, X) ~- D+ v3 i3 S3 z& l& FI12 / IZZZ5;
IZZZ5/ I12 ;
6 _" s  _9 Q( Y* z4 r; E! [- cZ * IZZZ5;
6 n* W. z9 A5 K3 d+ [& a( YI12 + IZZZ5;
. O6 A$ R! `" rIZZ15 meet IZZZ5;
$ }: @7 m- `6 i: p- dIZZ15 / IZZZ5;Z meet IZZZ5;
% N' D  G5 Y4 _! E6 r9 BI12 meet IZZZ5;
6 N6 H7 ~% F$ Q5 E- X6 ]IZZ15 meet IZZZ5;
3 z- ^$ k. X  T4 T# UIZZ15 / IZZZ5;

I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
1 W% ]( D/ Y8 Y% ^( A' TIZZZ5 subset IZZ15;
Integer Ring
( Y5 Z+ d; C3 f9 d# d6 R2 cIdeal 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
0 |3 e* s; ~8 E  _$ R( }1 EResidue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
; R! w& Q8 p. E; @Ideal of Integer Ring generated by 180
6 n* J* D& V7 c1 UIdeal of Integer Ring generated by 3
# G, |( p& a0 h  `3 PIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
' |1 y) M  ^, b! {& S2 UInteger Ring
9 ~3 L3 i  B7 @( yIdeal of Integer Ring generated by 60
  Y$ p, M1 P2 J/ J! R( {
$ y2 p% H) A# g# w$ Z>> I12 / IZZZ5;
       ^
" x% _. T6 |# {  `8 s" ^  RRuntime error in '/': Argument 2 must divide argument 1.

0 Y" |3 E4 z% }: Y
- ?% X% s: L9 Q) ~, L: @4 s5 G, X9 x4 p>> IZZZ5/ I12 ;
        ^
8 X3 B( {; O6 @Runtime error in '/': Argument 2 must divide argument 1.

# ~4 v# {3 }* c" r. `+ ]1 PIdeal of Integer Ring generated by 5
- H7 [0 M* A! I9 T! nInteger Ring
0 \  D) p% h7 K- Y" LIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
+ n  }  {3 @, L$ u+ c' \Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
, Z5 P5 F# V' F9 t. J8 _Ideal of Integer Ring generated by 5
7 u$ F3 ~7 ]9 f2 d+ [Ideal of Integer Ring generated by 60
1 L& y5 d& \5 Z  `' [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

false
true
true
false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
' h/ P: z8 a+ CI12;
ZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
IZZ15;
* M5 r4 z( o- o2 oResidueClassField(I12);
: Z$ H3 v0 Z+ Z9 e, c$ D! v! F- {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- M  D/ N1 c9 bloc< Z | 19> ;
loc< 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 | > ;

  P" H2 m6 ~+ Y& ?" ~: H+ N& mext< Z, 2 | > ;超越扩张到多元多项式
" k) y0 B% J0 V6 m
% o; J) F! T  gext< Z, 3 | > Completion(Z, I12) ;
: y" O; W$ X# K7 }* y) I4 q0 B& Y* U* f

8 `2 W" {; m4 ?# j% scomp<Z |I12  >;
0 v8 ^# F4 a: G+ a5 @$ ?. z    素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
1 s3 P# s( y& m! _5 y: \6 {4 U% i8 Xcomp<Z |0  >;

8 G, n8 L+ x7 H4 bInteger Ring
; e( d1 y8 n# p- w' @, AIdeal of Integer Ring generated by 13
+ V' u! K0 k# G1 f; o! E$ H% V+ P+ h! yResidue 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

% W. ^# d: P! E2 q>> ResidueClassField(IZZ15);
/ C, [7 M  f6 M" d                    ^
Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes

Valuation ring of Rational Field with generator 19
' ~1 [2 E9 I' z- AMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
+ j( X- }- M+ }' AValuation ring of Rational Field with generator 17
( @  {, \/ s" e% oMapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
0 a' p6 A/ v& e$ q% z4 bMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)

5 A7 ^- n9 D: `3 n) F>> ext< Z, 2 | > ;
      ^
5 V7 _  Q1 v- z" G/ K6 [Runtime error: This constructer is no longer supported
& J  G+ D8 g: |" r* A) i: {6 v, k. C

3 K# h$ T$ E& y! `# I>> ext< Z, 3 | >
. z7 U; f# o- v# P* t$ p      ^
Runtime error: This constructer is no longer supported

13-adic ring
- ]# v  U" I6 ]( ^$ gMapping from: RngInt: Z to pAdicRing(13)
# _, |" {; j8 k$ h1 B
2 R0 ]& a3 W7 C2 r8 W, V  E8 }% PCompletion(
    Z: Integer Ring,
" m' A$ K" J1 E7 I' Y! M    P: Ideal of Integer Ring generated by 0