一些初等函数：

lilianjie

:=IntegerRing() ;Z;
n := -1666666666234567890;
> n;
9 l- E- {. Q2 h2 s! V; b
> n:Hex;                           转16
7 [5 |% v) q  f7 H( `( x% A  dIntegerToString(n, 2);        转2
& a# p' r: u3 Z. A; ]# QIntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
3 o( L% P/ p; Q) i  ~; L7 C: O) _; L4 HIntegerToString(-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 Q$ M% L  |- @! A
m := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
6 t. a5 y* v6 ^k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
; b* ~4 C! w$ ?. ]  s2 i' w> k;
n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
% {% }8 y8 g* z7 D& Q> kk;
kk eq k;
4 C# V1 f- I2 ~' ~6 H; D4 r
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
# ^1 m5 }' L. X# {# x; {# g> k;
+ ], _# K  f9 _n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
6 V, V* j' j( Y1 M1 V3 l8 p: ~> kk;
6 @) I+ D5 N0 }  j0 @+ U7 {( okk eq k;
1 P1 m- \4 |; W( V
  z; p' k6 O' z* ^Eltseq(kk) ;Eltseq(-1/14);
4 r( Y% A3 L2 V" {( _



k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
1 X* b+ Q# r: g6 \% I" a6 ], \> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
0 a$ g! O  Z; W. Q
k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
/ A6 ?3 u9 e4 d' [1 [) K1 x" hn eq k;
/ E6 a( A0 X5 o! Tkk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
5 |; x/ F  {' M, i% @, c+ s/ O
3 O6 a8 g3 k1 {- w- hEltseq(kk) ;Eltseq(-1/14);
& M) o( g* k+ |, ^: S
- u2 B3 h! M4 M# E


. c$ j/ ~  O8 z6 u; N! }
& j% s7 F% Y7 V2 |
( r6 }7 L8 B3 t; v# ?% K  u

% l. E; N) Q+ p/ k
' a& a* C% w, Y+ n1 _5 E
* s$ c, y: @1 E8 e4 n=============
1 |/ f7 `1 a6 y6 H7 Y4 n6 X
+ g4 ^* }0 k4 O1 ~! R! ]6 V& u
1 T* D, H, k- {. q3 ]% G4 V) z
1 _* J9 l' H* w7 K) g/ H
- p6 e! l- u: PInteger Ring
-1666666666234567890
7 ]" [8 J) O0 Z& b# y5 ~-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
2 b" G0 N, c( q# j5 y) m-1666666666234567890
-1666666666234567890
0 J0 u" l/ f# H7 c0
) a5 X& @4 W, \* W' b  K# F# E1
5 V9 @' N. L2 n# ?- T. A0
+ g/ ^2 X5 j& R9 t5 I  t6 l9 ][ -1666666666234567890 ]
[ -1666666666234567890 ]
9 v3 P. K) o5 o# T5 G' T1
% z2 }4 V, @6 R2 c& S- }) x, y13
0 ~$ q# }! ?2 ?; @1

-1666666666234567890
true
-1666666666234567890
# g: p" b; B0 M- x, Wtrue
% M+ r3 K' o6 F! S) [9 F# H-1666666666234567890
; I) z7 Q/ k  `* E  Y8 j8 otrue
9 v7 \  [" a6 B7 T$ O7 J9 ^( M-1666666666234567890
0 `' k' j$ r# I% H2 q; atrue
. s" v. g: Z' W1 v; v' f[ -1666666666234567890 ]
[ -1/14 ]


) ]4 b7 @, _; S
$ f6 h$ W# s, a


- X7 O! U- a2 }
4 v" A  Z* N2 p* ^-1666666666234567890
-1666666666234567890
true
2 E8 i0 R; `2 n1 w8 Q-1666666666234567890
7 ]2 T2 M% j# c: b) ltrue
-1666666666234567890
true
; V! `( U+ m' i! l. E/ e  K+ z' k-1666666666234567890
true
[ -1666666666234567890 ]
( W1 n; l# p/ j. F: r[ -1/14 ]
/ Q1 j  q+ z0 A+ s# ^# U8 ~

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
s:=0x12345678111;ss;

sss:=Factorization(ss);sss;
" J- w! _3 \' ?1 j3 P- |) `. C6 C4 @sss1:=Factorisation(s);sss1;
) \/ g4 j: T6 u  ]+ z* \& h# I" AFactorizationToInteger(sss);
: Y; j9 C+ E0 l4 @$ x; iFactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
% O% z. W" I+ W8 E8 a+ HSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
: t4 ~1 d- a0 S. @6 j+ c2 G: S( \SequenceToInteger(ssss1, 17);转成2和17进制
- ^6 ~) r$ K9 K
12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
8 F7 Y: @, y9 }% m( f9 s5 x1250999894289
! S2 O0 b7 `+ m  ?5 @[ 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,
+ j( g, G+ c6 q( v: d: q& x, ~1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
7 p0 L7 ^+ l, R% @6 F* h) U; F[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
6 _3 E/ ]: K3 u! H8 x1 A3 x4 f1250999894289

lilianjie

+ H* J3 }; d$ _8 H+ _% ?Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
0 H- ?; L4 s7 @8 P7 X% Z7 T
" ~, Z6 c' R3 _2 U- C% ?
K:=Z!n1+Z!n2;K;
' N: D1 k$ P# m3 G
IsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
9 b1 }! P" ?4 k* z/ SIsOrdered(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。。。。
- ?! l" T& N. S3 C' n# \' l0 oIsPID(Z) ;主理想整环吗

) U; ~: W; h7 @4 HIsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
! z! d! B% o4 }3 z! T: J' vIsPrincipalIdealRing(Z) ;主理想整环吗
3 }* K! Q! s  g5 a1 ]: _IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
- q- S8 [* i1 x4 _0 v" I: [! c! CMultiplicativeGroup(Z);乘群
Category(Z) ;范畴Parent(Z) ;父环
( G! O% h* Z6 x4 Y7 ]' XPrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
  r* [* ^% _% L7 rAdditiveGroup(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
+ a- Z) D+ f+ Z. d* }/ f
2 i1 `* U5 D: b( O8 c, yZZ:=IntegerRing() ;ZZ;
% @/ ?" ^% Z  k: lClassGroup(ZZ) ;

===========

* U5 D  d. s5 K- _' G, JResidue class ring of integers modulo 5
1666666666234567890
1
1
5 ]9 Z) T2 ~9 T( S4 a( _2
9 u% Y2 e* [' ^) u8 o) i) D7 n$ Ctrue
! U9 X9 F# b! |9 m, |5
true 5
& L. h' Q& ]# y9 B' b4 \$ K( P7 Jtrue
false
1 `7 B1 T' ~- \3 B+ ktrue
5 C$ j6 J! b& h9 ~+ Q4 _true
true
* s5 T& B8 q1 C) Qtrue
true
true
; A' Q& N& w; y1 _2 d& utrue
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
2 ^$ g# c/ u; [Abelian Group isomorphic to Z/4
Defined on 1 generator
7 }7 @# j% X1 i7 P  YRelations:
    4*$.1 = 0
& S) p7 J5 O+ h1 [, {RngIntRes
# C. ^8 F5 b8 |6 S  ~Power Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
) `3 U( V) Y7 }9 uDefined on 1 generator
; D5 k2 x# Q5 hRelations:
' F  m- S( \( I7 I! b' s" M# w, D    5*$.1 = 0
* q$ F& s% L7 V# A: p
& U, r8 C6 w( H& ?3 d>> ClassGroup(Z) ;
             ^
3 e8 [% L/ x2 _0 J8 y* FRuntime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

Integer Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
' Z! |% @( K7 `7 M7 @6 E. {MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
# y; U$ K# P+ e' @
Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
: m% G' z# g, v$ c3 K( E" ?* LMultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
# O% J* r+ K! b1 u+ e


0 B) ~& w+ L4 D" s! UResidue class ring of integers modulo 12
/ k% Y& m( _# n) U+ E) aAbelian Group isomorphic to Z/2 + Z/2
; S9 ~9 P* V! i) MDefined on 2 generators
Relations:
    2*$.1 = 0
/ [. B" h, d7 Z% }$ Y& I; p' l3 [$ ?& e    2*$.2 = 0
Abelian Group isomorphic to Z/2 + Z/2非素数环的乘群同构两个小群的直积（1*11    5*7）Defined on 2 generators
2 m7 o0 x# C: E, Q8 v4 H) MRelations:    2*$.1 = 0
' t  V: o6 J& b    2*$.2 = 0
4 C) c! b1 y& Y: B9 sRngIntRes
8 U3 Y# F* }0 @1 W! [" S, O$ hResidue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
; t3 |( v# D: J; `1 ?    12*$.1 = 0
8 _+ C4 m( ^. b  F0 X! t4 KResidue class ring of integers modulo 13
' u, k  G# }0 p! A1 j4 P5 Y8 kAbelian Group isomorphic to Z/12
Defined on 1 generator
4 s( C0 M/ }; A8 a8 ERelations:
    12*$.1 = 0
. F( b& w0 j" rAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
* a% ]& ?0 M2 e3 x# [# fRelations:
+ O2 B9 O4 O1 V$ y8 Z+ X    12*$.1 = 0
RngIntRes
: i4 S0 K( d+ c. JResidue class ring of integers modulo 13
$ E0 U6 i6 O9 L* s! K# YAbelian Group isomorphic to Z/13
% a% B1 C5 @0 u" [' G1 E/ ?Defined on 1 generator
6 h  ?1 j! B3 E: ^7 zRelations:
% V( y  j' ~- S+ \0 |8 {# a4 q6 x    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   

R:=IntegerRing(12) ;R;   
/ v7 J, J8 R4 Z' Q' k4 @: ZS:=IntegerRing(13) ;S;   


PrimeRing(R) ;
Centre(R) ;
* k5 O6 p( Y7 m. z
Characteristic(R) ;
3 X+ s# z0 N! I# R ;阶----元素数
% S& C1 ^- d5 ]" A8 f0 V! Q# NIsPID(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) ;
9 y9 @' S6 D. lHas**(R) ;

IsPID(S) ;
IsDomain(S) ;
Has**(S) ;
2 P! [; g/ b  l  B; V! N1 z" ]; Y) nR eq S ;
R ne S ;
  {, C' b2 v2 i5 G6 l
/ ~( q* Y$ u6 b6 P4 l6 I% XParent(R!123) arent(S!123) ;
Category(R!234) ;Category(S!234) ;

; w+ [/ m( _# z0 o9 Aa:=Random(R) ;a;b:=Random(S) ;b;
' D) U9 O( q+ l# G( e8 K+ Z1 g% oRepresentative(R) ;
9 V; t2 U4 D3 h4 D( ?Representative(S) ;

3 I+ U) q, M) K; z(R!a) in R ;
+ P1 k% ^( u3 a( i3 B2 ]" `( l(S!b) notin S ;
6 ?+ B8 J9 j" e+ N1 C5 xIsUnit(a) ;                是单位吗
5 Z' O/ j7 d1 v  C3 aIsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
& R6 U: L) i; p# v" ^- aIsZeroDivisor(a) ;可除零吗
) R- s' }4 T  @6 ^0 CIsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
/ N' @+ V$ R1 N: F3 v: p' B
6 c; O5 H- }* p( L* P5 ?Z!a gt Z!b ;
, q5 Q4 y0 _3 |! W, P- {+ k) pZ!a ge Z!b ;
Z!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) ;
* u5 w# d  f# q, a, X: DMinimum(Z) ;

Maximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;
0 {& X! u* p3 ^* j

* w# M% h) h" V% ?- b6 Z
Integer Ring
. E) p1 X  s6 M) r2 LResidue class ring of integers modulo 12
. G& C: q3 A. K4 RResidue class ring of integers modulo 13
) Z* Q/ Z" z) A1 e# B; }% jResidue class ring of integers modulo 12
' K$ _) J: O% ^' {$ N0 lResidue class ring of integers modulo 12
, V7 N2 h. }" E12
12
2 }  G/ |/ |' H2 `false
. Z* M! D$ z; qfalse
true
' K1 e5 k2 v% c+ R  Ttrue
* E4 e7 G. k6 x  N' J6 ftrue
true
5 C; R1 o2 p) h5 L6 hfalse
& r/ Y) R3 l6 u: A. ^/ f' {true
9 C3 L; x6 E* g% @* w  P9 zResidue class ring of integers modulo 12
' @6 G9 N9 P/ Z, N0 n; [' q/ \/ vResidue class ring of integers modulo 13
. E+ a0 ^8 D# eRngIntResElt
RngIntResElt
9
12
0
0
true
false
false
0 E* }) `% I8 ~1 m) B9 qtrue
- J* O& `7 J6 r0 {2 b; afalse
true
# n/ {6 P0 e5 u  `- H2 yfalse
* I; {- J" J- b0 I5 ?2 R2 kfalse
false
7 g. Q$ S" Z. U; u  F/ Dfalse
true
9 w% R2 u/ r: O0 Vtrue
12
2 F( ^8 Z" h+ [% F1

# S: N( m2 |! C3 t" A>> Maximum(S) ;
          ^
Runtime error in 'Maximum': Bad argument types
6 C1 `! {. T' ^& y" H3 C# L/ C" rArgument types given: RngIntRes

9
) r9 n4 @" u6 E, }8 n# C; M3 G
>> Minimum(R) ;
          ^
Runtime error in 'Minimum': Bad argument types
. Z4 m( M4 `" }) W3 XArgument types given: RngIntRes

lilianjie1

' @$ x, A; R4 G+ R
+ x7 T) E; M, a9 tZ:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
ZZ:=IntegerRing(15) ;ZZ;   
7 x" I3 s/ B# ^* H# m! UIZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
; h2 C; F1 T( c2 [7 a4 ~* L9 z! JZZZ:=IntegerRing(5) ;ZZ;   
/ G5 O* l3 d7 n) Y, @, jIZZZ5:=ideal< Z | 5 >;
IZZZ5;

I12 *  IZZ15;            理想和/积/并/交，
; v% ^: y- C/ [# o2 m3 e$ @2 x7 y( Q& ~理想和是理想对应两（可多个）元素加，
; p: n# N+ j( I% I% J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，
理想交是理想（可多个）元素交，理想交一定还是理想，
6 t* U% [9 g/ K/ ~4 t/ l+ h
理想积是理想交的真子集，极大理想交是理想------J根
, f0 I) c4 P% J; P. W理想商就理想间同态：是必须能整除
$ w! j. `/ J  e/ ?1 r2 Q% r$ i; U: PI12 +  IZZ15;
I12 meet  IZZ15;
, U3 |+ E  O1 a7 }$ q6 N
I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
I12 / IZZZ5;
8 ?  M6 j) k" t! RIZZZ5/ I12 ;
( k8 S* R; a4 P" T1 Y& wZ * IZZZ5;
" q9 p/ @" o1 |3 U$ c- `I12 + IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
* J% Y) b! D5 I/ e. U! }; eIZZ15 meet IZZZ5;
7 j" J) V8 O5 K. |; p* `/ D9 mIZZ15 / IZZZ5;

6 N: G: `- F4 X6 O( g; _I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
/ h1 t2 T# G% m. gIZZ15 subset IZZZ5;
IZZZ5 subset IZZ15;
Integer Ring
) A1 I; {+ t9 b6 F. K& E! o6 H  g/ tIdeal of Integer Ring generated by 12
% @: N& v" X$ r* R  bResidue class ring of integers modulo 15
" s( y- W+ w7 c6 GIdeal of Integer Ring generated by 15
false
Residue class ring of integers modulo 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
: L# w, x* W4 q6 o7 x/ H' dIdeal 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
- l% o& D3 q1 f
>> I12 / IZZZ5;
! R% Z* q7 @% f       ^
2 p8 @$ `7 r/ ^5 F" n) d" nRuntime error in '/': Argument 2 must divide argument 1.
$ @8 v3 M0 W2 w0 C/ V; D
8 o; B0 f$ g( [9 D% n7 `& c$ C
>> IZZZ5/ I12 ;
        ^
Runtime error in '/': Argument 2 must divide argument 1.
) I, X+ m  R+ ^: L: l, M' t
1 }8 ~8 H& U; z: g; s3 qIdeal of Integer Ring generated by 5
Integer Ring
# h, i( u0 `1 d' wIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
0 U5 S1 T+ H7 p+ U1 ]Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
Ideal of Integer Ring generated by 5
9 l: x2 a8 Q* Y! A" ^3 ~6 ^; DIdeal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
6 p* ?- @9 q3 v: ^" uIdeal of Integer Ring generated by 3
- j8 x: L2 M1 d4 b* t8 BMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
: _& n# f% B( D- W7 Z7 t- Z; V+ O
false
true
true
false

lilianjie1

Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 13 >;
) B: k! l' z* {: eI12;
$ q: L8 w! i, }8 D% g" E; zZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
; D. B" o( L9 q  r! Z& I" g8 ~IZZ15;
ResidueClassField(I12);
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
loc< 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 | > ;
% x6 U# W- M$ O& Q* s, F/ q
8 w, _2 h1 m" J2 \" ], bext< Z, 2 | > ;超越扩张到多元多项式

ext< Z, 3 | > Completion(Z, I12) ;
( l) K% }# J  A; U0 s

comp<Z |I12  >;
# S2 c9 l2 Q8 i9 K" x% b    素理想零理想完备化，和P进环联系起来
; q5 r: b4 F: |+ f3 P/ tCompletion(Z, 0) ;
comp<Z |0  >;
, z% @2 B1 T, w/ p- R9 k! e5 D: ]0 N
& N) H) u- ~4 K' kInteger Ring
Ideal of Integer Ring generated by 13
% r1 `7 ^7 K& R3 |$ NResidue class ring of integers modulo 60
7 i, c- e2 O6 y7 R  [7 L: w" GResidue class ring of integers modulo 60
Finite field of size 13
- P7 e6 {" u* I7 I0 c( SMapping from: RngInt: Z to GF(13)
+ W8 W! Y( s( y- i# v1 V3 mmodulo 13

>> ResidueClassField(IZZ15);
8 H( |+ W2 m8 N! X# Z" V                    ^
Runtime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes
0 R: N6 h, N0 C
: y$ Z8 X! ~5 ]; NValuation ring of Rational Field with generator 19
- D/ J5 X( J# o' r, L& vMapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
: l8 ?% J' h1 Q2 {  @5 SValuation ring of Rational Field with generator 17
. m5 W! g/ S4 o, b4 Y5 XMapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
9 Y* `# g3 ?. U  O; 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
1 M6 {" m  z; j7 O& J) b1 r7 WUnivariate Polynomial Ring over IntegerRing(60)
5 k" e$ u3 l1 \% X/ K4 a
- s3 R% T- k) d1 V>> ext< Z, 2 | > ;
0 p8 c: G# M7 n$ D! N      ^
  e1 U4 f) {* N* K5 IRuntime error: This constructer is no longer supported
7 Q. L( X: _' x7 m3 u

& |% h7 S* v* {; \' V>> ext< Z, 3 | >
      ^
Runtime error: This constructer is no longer supported
  {4 s# }) D- l1 V) b
/ R0 B# {0 `- y13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
7 U/ o: v% Z" m; Z8 b6 b; `' X
Completion(
! Q+ v( u/ J! p2 V- Q    Z: Integer Ring,
/ N* V, ]' ?3 l5 s8 r1 K4 O+ G    P: Ideal of Integer Ring generated by 0
' I) j) g) Z2 K0 ?/ [. p