一些初等函数：

lilianjie

:=IntegerRing() ;Z;
  K# k+ g- X: l# M- `9 R0 g- ]n := -1666666666234567890;
> n;
8 E" S% m# e9 ~. S
> n:Hex;                           转16
IntegerToString(n, 2);        转2
/ R5 p6 k& H  O0 Q* z2 q; XIntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
7 Z1 L  K& L, j4 c1 v  C, c- [Identity(Z);                  
" V# b( u$ F2 v/ i. h3 QRepresentative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
) t0 ?! Y2 [: F' q% l" o; ]9 V
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>;
+ \! d) D, J- P, x6 ?> k;
4 W. i# f+ I; ~# V# Bn eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
5 b8 a7 L7 F5 H0 x0 R9 J> kk;
kk eq k;

k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
" ?% Z; U* I/ j1 A2 m# Tn eq k;
$ ~4 q, {7 p' Q! dkk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
& O$ s/ A" g% H0 I> kk;
kk eq k;

  W8 ?8 F& w- `5 h$ {Eltseq(kk) ;Eltseq(-1/14);




- q9 d! z4 |8 M: q  D4 {k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
$ w/ r3 Z8 t5 P5 f- ?> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
2 q0 w! Q8 [: y9 N> kk;
5 `% n# r. M/ J- f6 d0 lkk eq k;

" p0 e* W; F. M* Gk := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
3 I0 H: E# p# t> k;
! ^! ]$ C2 E1 N- _3 c9 E2 s9 jn eq k;
% p" V; M5 o. b& y$ {" p" k' qkk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
: p2 A1 l6 |  x
Eltseq(kk) ;Eltseq(-1/14);





, d9 ]7 f" b) Q5 q6 ?  v


5 S" ^* b' p% C
" V( y8 ^5 ]* B" f
=============



9 i" t/ e# n# y1 s
Integer Ring
-1666666666234567890
, Z+ a+ A; h3 P" O/ ]7 b-0x17213080A7E55CD2
) x* u2 V- E9 M- L2 Q-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
1 i# {# i; A' m/ ^" @-17213080A7E55CD2
# ?, T& V' U! S% R4 t-CNUO0WGPY9CI
-1666666666234567890
' T( Q* J( p7 `& T8 e-1666666666234567890
5 J" M: I) X( ^8 V. b0
1
0
# I+ g" ~/ T4 l/ c8 n7 R( q# T[ -1666666666234567890 ]
1 I" v2 s, s/ {6 h2 d[ -1666666666234567890 ]
* h5 u) i$ N: I/ P1
% R* T+ p( R3 e( r4 D9 g13
1
5 ?7 X2 J: d/ O" v+ z" u# Y
-1666666666234567890
# V$ Q, C2 w1 I6 o# g! Ztrue
5 c) X! Q; g9 _$ H7 H5 w" }-1666666666234567890
  D/ x6 c& d, a0 j( [& Qtrue
-1666666666234567890
, ^7 O; b  e% j% ]. @1 S& n1 t1 [true
5 t9 W. R( R$ H/ T-1666666666234567890
$ [' ]+ L6 e4 B- P& ]true
7 |) |6 c5 \) E. \* f[ -1666666666234567890 ]
[ -1/14 ]
6 _9 G0 \  B6 e. U; G; |8 m; A2 ^
' F& ^/ I4 o# K: T6 x7 d



. a5 A+ v3 T5 z/ t# u" m- v& S+ r
9 L- [6 J5 T4 h2 L& v7 S& N
- t) n  ]3 U* u8 V-1666666666234567890
-1666666666234567890
: y* Z/ E1 y2 k# G% H( |1 @true
1 v: h* m- x2 G# V6 ~& A& {-1666666666234567890
" T& x3 t4 p! D( G" g- Rtrue
3 G& C1 B$ V# j, ?9 G! a( i( X2 M-1666666666234567890
* e  K8 |+ E( b1 b) X6 o# Utrue
-1666666666234567890
4 V/ D" A: \! [4 K6 ^% @: g6 D6 Atrue
, j1 |6 }8 y+ p' u0 |% M' p+ S3 Z[ -1666666666234567890 ]
[ -1/14 ]

# W: c- @) A- Q$ @: g

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
s:=0x12345678111;ss;
" y% E$ z' A. d. l9 C$ J
sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
4 S1 B  P0 N) k) UFactorisationToInteger(sss1) ;
" L- d5 [4 q$ J9 jFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
  p: X  T; R: H2 y7 f$ h% |SequenceToInteger(ssss1, 17);转成2和17进制

12345678111
3 O! q3 d* M; S0 ^, Y$ f. d12345678111
- R/ W" c. d% b% p+ g& A, @[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
2 Z# d' H+ ~+ Y1 L$ U1250999894289
1250999894289
- y5 w- H7 r% s2 p[ 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 ]
" O3 T, N+ H( q( m2 X12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
+ K+ {2 L; v  |' ~2 z, M12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
7 l! K' Z+ K: V" G1250999894289

lilianjie

1 p) @0 i; I3 M0 @2 [Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
4 G/ S- l1 W0 ^; ~& G> n;
n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
1 u- f5 E- J9 c& Z* g$ [
- {8 f4 u# }1 E) ]- D7 T
K:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
; R2 t! R) b" ^) WIsFinite(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
, }* S( W! [; [# m7 d
IsUFD(Z) ;唯一分解吗
' @. C! ], i: i* ?- m# zIsDivisionRing(Z) ;除环吗
: G$ c8 T, h5 Z, F/ \$ WIsEuclideanRing(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
7 ~* {9 s/ |- w! v( ]: Y2 BMultiplicativeGroup(Z);乘群
Category(Z) ;范畴Parent(Z) ;父环
, S% ~' }" Y( |2 n" D; P: l" SPrimeRing(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

' b0 L) a5 H# v5 R# H, @1 ZZZ:=IntegerRing() ;ZZ;
: m; H# f8 V  p" Q! HClassGroup(ZZ) ;
, P+ J+ u' V* C7 V0 K& S/ I
2 ?3 U: ~- P4 A; a7 w===========

- h5 g/ V( L# i$ |4 U) ?! AResidue class ring of integers modulo 5
1666666666234567890
' g6 ?: B; e* ?- }  d+ H) {1
1
2
8 c0 w/ D. W: q# [. atrue
, o2 G2 G. u: G% g0 r6 K0 i7 m5
7 B) s1 ?# ]+ K9 ^* m* m* G2 ltrue 5
true
# v5 T2 k3 @: C. `9 Q5 k# Gfalse
0 U' m8 S/ k) Z, y, N5 G! d& B( otrue
7 k9 Z( ]4 N* d, _. w% _* K/ Mtrue
true
true
true
true
true
; |3 V4 a6 }0 H  ?/ B# X, iResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
. j- m$ M, f" }! Y2 Y; f1 P, c8 T0 `Relations:
/ d/ L8 O% W" w- b% _9 m) o2 }    4*$.1 = 0
Abelian Group isomorphic to Z/4
) T! |4 ?6 p6 S% NDefined on 1 generator
8 d' u; A4 D9 F( J- z0 Z3 Q4 E8 B* SRelations:
) K7 b; }+ j. T4 f: K  M    4*$.1 = 0
: W6 q1 K( X/ b: ?7 |RngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
) ]$ W! W$ }7 Z; i2 YAbelian Group isomorphic to Z/5
+ i  ]9 o( c1 i6 d! |% tDefined on 1 generator
9 m  G3 |+ a% a, N! A6 E1 bRelations:
/ S; i' M' M/ e    5*$.1 = 0

>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
# P- Y! j5 g- J0 }" X( A* e% UArgument types given: RngIntRes

Integer Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
( ]0 g% M& R6 p4 k. y+ | UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
8 Q3 R( h" Q* b! s& ZAdditiveGroup(Z) ;

5 d1 o, J; I: i  a; C! s) BZ:=IntegerRing(13) ;Z;   
3 o( ]& i$ t& n* N* _! s% [ UnitGroup(Z);
* B9 E. A5 N! {  t$ K" |MultiplicativeGroup(Z);
$ J; z4 I- u! U/ l3 h- ?( U/ MCategory(Z) ;
9 o( h2 z8 P/ V! |8 {PrimeRing(Z);
5 f7 t9 I8 h8 g( R7 c3 R; IAdditiveGroup(Z) ;
# m4 k" Y7 Z! a4 `: j6 V
* l- S# R: \3 V' ^$ q
9 t: o/ ^' D' c" y  T
Residue class ring of integers modulo 12
) u* _% ]0 C+ ^6 q& ~6 E4 K+ jAbelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
# }- M1 ?; `, `7 u0 l* _8 eRelations:
    2*$.1 = 0
  _& j& c, q% m& O7 J5 j- [    2*$.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*$.1 = 0
8 y8 e/ c) z8 Q    2*$.2 = 0
RngIntRes
4 E6 l$ t0 r4 W- K2 b5 p- GResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
6 }1 y4 u. B" @) pDefined on 1 generator
Relations:
7 d+ B6 k# R. C. e2 Y    12*$.1 = 0
" C* D; [* I7 fResidue class ring of integers modulo 13
* ?# g; ?* C  r2 k0 n# V: I; KAbelian Group isomorphic to Z/12
% o% j) }6 t1 {$ y- u# |Defined on 1 generator
( H& e0 [) p! iRelations:
    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
8 _* l6 W( k7 f1 e; x    12*$.1 = 0
1 |/ A3 t! F, oRngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
Defined on 1 generator
Relations:
    13*$.1 = 0

lilianjie

6 y2 T8 f. `+ ~+ I! V
: g" A0 M& D8 V8 B4 r" SZ:=IntegerRing() ;Z;   
; t# B! ?5 z$ [3 ?8 @0 M9 ^7 K+ k
* w# Q5 I: Q* G% BR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   
4 r$ [8 V  A" t. I* ~

PrimeRing(R) ;
Centre(R) ;

( v* {( h* H6 z7 ACharacteristic(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) ;
* X1 G/ m- d: eIsDomain(S) ;
Has**(S) ;
2 K. \" A' n* q- Y* SR eq S ;
R ne S ;
8 |$ ?! d4 x' A+ i" z5 w! K) g1 a
Parent(R!123) arent(S!123) ;
# |9 N; S  u% Z4 p) C: s5 S" ACategory(R!234) ;Category(S!234) ;

a:=Random(R) ;a;b:=Random(S) ;b;
0 S+ v+ v2 p% [+ mRepresentative(R) ;
Representative(S) ;
3 I, C( l! Z. @- d' c0 E" C
- u( e9 Z# M4 Q(R!a) in R ;
(S!b) notin S ;
IsUnit(a) ;                是单位吗
4 N+ ^- `) L* k2 a* @IsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
' \$ w- F* k0 IIsZeroDivisor(a) ;可除零吗
# a# T: i1 W9 P# u7 [4 U- wIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
; w: {; a) \# [6 n- e
( \2 r* o* q0 |8 D& J: }8 DZ!a gt Z!b ;
! x# @( l$ D9 L+ n+ k/ VZ!a ge Z!b ;
; T- Z& i4 `8 F* J( ?, HZ!a lt Z!b ;
/ L6 Q; \& x) g1 j3 ]& DZ!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
Minimum(Z) ;
  l% \) u5 O! Y
Maximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;
6 C( }8 }% m5 A. z, Y$ V


Integer Ring
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
Residue class ring of integers modulo 12
2 M8 W+ G( _/ l' F; QResidue class ring of integers modulo 12
! N8 `3 K) U0 L4 |12
& n) D1 N) j4 Y* K5 S( B12
9 D3 g+ g' j4 R; a" Ifalse
false
* d8 _9 S8 k# ]+ H( w! Etrue
true
, C1 t) u) J8 [8 [( M/ \true
: b/ x5 t( L+ K- z! V; ?true
) S6 j( ?( H) s/ y. J6 Pfalse
true
8 P& y( X2 v0 ~5 iResidue class ring of integers modulo 12
, l1 N0 g; e: m+ K6 x( GResidue class ring of integers modulo 13
7 L4 ?8 J( |) x3 [2 w+ g4 Z; {RngIntResElt
' @* g0 E- X3 BRngIntResElt
5 y4 K0 j) R8 R( c% Z( V, |: _% v9
12
0
$ G& i/ @# T$ b/ V' A: o* E3 _$ l( t0
true
& R3 h# ~0 ~, j( j3 l$ k  t2 M' Afalse
$ k$ q! \9 _1 C' c2 @false
/ @" I2 m/ x' W4 o' R2 |true
false
true
( d4 {2 P( k7 O3 xfalse
. M" Z6 y" r+ b  c( ?; C+ ]false
' l' {" v- G; p( bfalse
false
true
true
12
1

- n# _' \; Q4 x# M/ T3 h>> Maximum(S) ;
+ u/ E+ c8 ~) _* i6 O/ O          ^
$ T4 I$ s' d* @& |Runtime error in 'Maximum': Bad argument types
# F  p: o1 m: Q  n0 Y6 uArgument types given: RngIntRes
* Q7 \) W( d2 P1 {& k
$ {# O: h0 K; K8 N8 _( p9 z8 r. J  d9

" t8 }3 b: h; C: g! [/ ^1 y>> Minimum(R) ;
          ^
& |( W% J) h$ R( ~% [Runtime error in 'Minimum': Bad argument types
6 W9 u3 q* D* o, N# \5 F( R. fArgument types given: RngIntRes

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
1 G8 S3 i# m) v( W& u  f3 JZZ:=IntegerRing(15) ;ZZ;   
& U: d1 x1 J2 ]4 a/ n, |: t  aIZZ15:=ideal< Z | 15 >;
, Z; P5 H# n, p3 M2 r. {IZZ15;
3 {! T  i) R: I' J, Y; ]I12 eq IZZ15;
; k/ F- s2 Q. }! B/ j: u  J7 ?Q1:=quo< Z | 12 >;Q1;
ZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
: h% N" j) W8 j  T! QIZZZ5;
+ Y/ V( ~9 s! f$ S
0 j& p: x; t7 h: w( yI12 *  IZZ15;            理想和/积/并/交，
& V3 P) H9 |+ s/ w$ Q8 g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，
5 W! l, m; V9 m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，

* u0 z+ q) V) D# T, ^7 [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
I12 +  IZZ15;
. ^* X$ S. B+ x; II12 meet  IZZ15;
# a( i& H: z/ h
I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
I12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
. ^: ?. K  _: _& V0 \6 bI12 + IZZZ5;
3 ?/ C) K) @, L! wIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
5 J2 K2 [/ I2 V, H, OIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;
+ a  s4 e5 X% q" Q  e, w/ y
I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
5 W9 X9 s  i7 ?+ f1 N2 u; a' }6 MIZZZ5 subset IZZ15;
6 C; L. p8 ]) I2 P% YInteger Ring
9 V# j$ b% h7 B8 A$ I6 CIdeal of Integer Ring generated by 12
( V- Y: w/ }" r7 W3 g0 UResidue class ring of integers modulo 15
5 K; A. J1 `8 C$ s' w  JIdeal of Integer Ring generated by 15
false
Residue class ring of integers modulo 12
6 t8 c% |3 I2 K; r. |4 t9 TResidue class ring of integers modulo 15
, }5 X# g- l2 }' G3 {: i  }Ideal of Integer Ring generated by 5
3 J# U9 f) T9 P) N  w4 TIdeal of Integer Ring generated by 180
" |4 }" Q. T( t3 TIdeal of Integer Ring generated by 3
9 c/ P7 z* Z( F! j& AIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
Integer Ring
Ideal of Integer Ring generated by 60

; [( P! o3 _/ h$ \4 R" O1 Q>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.
9 N6 r& F: J& ?, i
) U0 k, W' z( O' m( D
>> IZZZ5/ I12 ;
        ^
Runtime error in '/': Argument 2 must divide argument 1.

: \; M5 t; G: }2 JIdeal of Integer Ring generated by 5
Integer Ring
( n. N! }% P5 ?, K2 {; K* q8 p! tIdeal 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
5 Y9 i% _* ]  Q) b5 D# QIdeal 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
& Y! u' _, y3 O9 _' q2 |) GMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z

false
true
true
false

lilianjie1

Z:=IntegerRing() ;Z;   
5 H2 ]$ M( K- Q' `. j9 e  JI12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
/ w4 y$ r. U; P; d2 y& b5 }! \IZZ15;
( Q/ q6 ?+ j' s( C5 @, j5 y3 AResidueClassField(I12);
) p4 ~9 S- K  }7 d8 O) W; x' EResidueClassField(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> ;
loc< Z | 17> ;
3 g+ }: B; O7 q4 p$ |- G$ w  `loc< Z | 131> ;局部化：一个素理想到原环元素的映射
& p; \! g2 D$ a# Vext< Z | > ;超越扩张到一元多项式
) C7 N  ?( N- ^$ c6 [8 R5 @$ }ext< ZZ | > ;

ext< Z, 2 | > ;超越扩张到多元多项式
9 s, O5 T% _1 D) n- N+ |+ T4 [
ext< Z, 3 | > Completion(Z, I12) ;


comp<Z |I12  >;
- c. F$ q( u9 W    素理想零理想完备化，和P进环联系起来
  y3 r, \7 W% K* QCompletion(Z, 0) ;
comp<Z |0  >;
7 c7 i5 s0 O4 g2 `2 u
: u' x/ x3 `4 W- s# aInteger Ring
7 @/ a8 P1 _: `* p: pIdeal 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
) Q. z: }6 e6 m! C4 nMapping from: RngInt: Z to GF(13)
modulo 13

>> ResidueClassField(IZZ15);
$ c# z) e4 a; K0 Z                    ^
Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes

Valuation ring of Rational Field with generator 19
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
4 G, O# r/ T2 a/ f4 aValuation ring of Rational Field with generator 17
6 Q; r! \6 ?+ ~Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
$ ?, [; k9 b, D6 j" @* p% v3 SMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)

>> ext< Z, 2 | > ;
      ^
Runtime error: This constructer is no longer supported


>> ext< Z, 3 | >
      ^
Runtime error: This constructer is no longer supported

+ f2 E7 l2 _4 v13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
  {! P  w' N/ a; g
- m- n- O! m- ?Completion(
    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0