一些初等函数：

lilianjie

:=IntegerRing() ;Z;
. `# q1 ?: g$ {: vn := -1666666666234567890;
. c: h: H8 F( `. \6 ~) X+ g# ~% i> n;

> n:Hex;                           转16
IntegerToString(n, 2);        转2
' d& A' n" Q4 h& v; b5 yIntegerToString(n, 10);       转10
+ a* w) ]% }  ?9 t$ VIntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
Identity(Z);                  
  }# F) ~" R" K, X: ERepresentative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
6 t9 \' {% e/ }5 ~7 m' d
+ z1 o  s, i+ K3 mm := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
: {! i  c1 C& _5 @* D, j# ^k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
: S& D/ {+ H. C7 Sn eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
3 }; B$ e  a8 @4 p> kk;
kk eq k;

, B: L8 ~3 u$ H8 R& _# {. A- }; Ck := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
0 [: }% }: |+ z6 f> k;
4 [- b3 X# I& i* {: j6 ]: k- X1 Qn eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
6 l( d& A( k# ~5 G5 bkk eq k;

Eltseq(kk) ;Eltseq(-1/14);
' }( {% p0 L  t, h  C" P# W
) Z- j# ?# P4 L1 t

6 p" `7 i& a/ v, Y7 h
$ U3 W2 D' Q! n! E' pk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
) R: \. v- e3 O> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
( f- C" K3 b: J9 z. O+ ]kk eq k;
4 @! {5 m$ L  I
1 f$ Y# f. `: }* Xk := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
, F, Q! Y" w' g& h> k;
n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
$ Y& N' Z; m& k
/ e9 e) Z& t( ?1 LEltseq(kk) ;Eltseq(-1/14);

' d! H+ K/ W8 p4 a


; h" f3 T( m6 y
; I' q: _8 r6 X1 o! p
1 Y4 |5 g& Y3 @1 K  w1 t
1 s( Q2 e0 O1 L: E3 y' ?
: h: m! G9 D% |& A1 Q& |! ^

) X6 f% s3 L9 H8 ]! P6 m=============

3 s/ ^# @+ S" K+ z: {


9 B1 H1 N! p' R2 G1 UInteger Ring
-1666666666234567890
  Z) L3 X" M, f5 C4 [-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
* i" H! c$ |0 a, U( p-1666666666234567890
, N( o  v+ [3 s7 u0 D9 V# s-17213080A7E55CD2
-CNUO0WGPY9CI
; ^/ a' o% Y$ n+ L/ r1 F-1666666666234567890
" j# O1 X3 B: i-1666666666234567890
0
1 @  ?7 ~- X% y2 x1
0
5 F7 v1 a) B* T- S# ][ -1666666666234567890 ]
[ -1666666666234567890 ]
. h& [0 B1 c9 u. V" c% ^, d1
2 P2 |7 p% X: c, F* R% Q/ f13
4 k, x1 v, w# z$ _4 V  A1
- E5 K# b% G3 Q" Y0 S% R$ x
) Q6 ]: K  z  ^-1666666666234567890
" d: G$ R; S9 C/ D- d* Ztrue
8 v8 d0 y2 x4 C; O, B* E! I-1666666666234567890
true
  U: X1 b" a# g, [( x% x-1666666666234567890
true
-1666666666234567890
true
[ -1666666666234567890 ]
2 v/ D' r  u! C+ G  B[ -1/14 ]
# u2 k( F( y6 L, h2 s  @6 J
: i+ s/ f0 h& X! Y# L( J
: M, F! l% V/ E; @2 F" j! n3 G
: s# j0 Y, |  Y8 y8 [

# E6 M7 a- ~; l) m" Z# T1 s3 U
- I! J. h- n3 O
% P2 Q: m! Y$ i( V; W( [-1666666666234567890
3 b$ L3 M+ k/ Y% R2 k" S. \-1666666666234567890
. P! F1 U( U# S1 {true
-1666666666234567890
  L( h% f' K- x: ~1 K$ a2 btrue
-1666666666234567890
, t% m/ s& ]$ r% v- g, ?( Z0 A6 qtrue
-1666666666234567890
true
! Y8 U2 m) b! K2 ^1 i0 X[ -1666666666234567890 ]
[ -1/14 ]

5 i6 C3 `( d( e: l' s  t

孤寂冷逍遥

lilianjie

ss:=12345678111;ss;
$ v8 ?; f% e4 L0 m, k: Is:=0x12345678111;ss;

sss:=Factorization(ss);sss;
0 z& }1 \6 z( V- usss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
" q% t4 w' i) \  z( M8 n0 l; Z, ?/ {FactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
- l2 S7 l2 j/ xssss:=Intseq(ss, 17);ssss;
9 X, C$ I$ r3 E: A) |SequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制
! {, P8 p  u) j& T8 h- I
12345678111
" o0 n! x1 J9 Q% F12345678111
2 N, t6 N8 t% ~4 S  h[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
' K/ `, P; |8 ^6 r% [5 S) b4 k12345678111
! `# a9 H7 A2 O1250999894289
1250999894289
. p4 W3 w' L, z* K) K1 v  V" B[ 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,
. s( b4 \; _3 X; \$ t1, 1, 1, 0, 1, 1, 0, 1 ]
9 _  {: @; Y/ \12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
- p: S8 b# d' x6 G; Y  T0 N12345678111
! x* Z# x$ z# T- L1 e[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
2 \+ P3 x1 X& j; d, X1 U9 n% V1250999894289

lilianjie

Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
6 R& N& P& E: D1 V> n;
n1:=Z!1111111111111111111111;n1;
1 W6 i7 b' P& u( }n2:=Z!11333331111111111111111;n2;
. @0 i$ ?; {) E2 }. R  \3 J3 z4 \3 g

* Z# O" ~6 v( zK:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
; U  W( `: O1 w3 R3 OIsFinite(Z);有限环吗
7 p5 H1 D8 r$ }% L' p4 U: k) X% x: NIsCommutative(Z);可换吗
3 i( q9 k, B  f& X3 f2 MIsOrdered(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。。。。
7 J$ W, |; G: h2 G* ?3 kIsPID(Z) ;主理想整环吗

IsUFD(Z) ;唯一分解吗
. |! L5 @6 `  i9 W+ aIsDivisionRing(Z) ;除环吗
; O3 @2 b$ |8 k2 u' w: HIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
5 q8 P1 o; }% N7 J# t$ BIsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
! Q9 K) a& u5 n. g2 h* {MultiplicativeGroup(Z);乘群
5 S! C" E! h' O: V7 wCategory(Z) ;范畴Parent(Z) ;父环
6 F. p6 l4 H/ P! c; gPrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
3 F* }1 T8 c7 O# y7 VAdditiveGroup(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
6 d3 V* \/ f( S, y
+ e0 K, h+ o; X, J  ^1 XZZ:=IntegerRing() ;ZZ;
: Z, `$ `. v6 L, u. m6 j. kClassGroup(ZZ) ;

5 O! v( |5 N0 a" ?. u0 M. w# L& i5 H===========

Residue class ring of integers modulo 5
. ]" n8 a) f/ k9 f5 T: J; J1666666666234567890
1
1
: r7 N4 y: H" k2 Z9 T2
true
  j- Y  r8 \* f2 W; ~( Q3 [. O( {5
2 i0 X) \  W1 G5 R" p* P7 etrue 5
true
false
" {: T* \) z' L) rtrue
6 Z3 m/ c  C7 `: i5 y1 n& h( Xtrue
true
true
true
true
true
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
, S- ]! {" P  C8 w& `' }8 dDefined on 1 generator
Relations:
8 W5 c* l* p4 p" ~    4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
6 `$ ]3 n/ o, k$ p# \3 @Relations:
% `. v; q' K1 g$ V& `/ |, I$ H: _    4*$.1 = 0
' W! a) q7 x0 F% ~' V* I) LRngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
8 G% O7 }) K/ A4 F0 fAbelian Group isomorphic to Z/5
Defined on 1 generator
1 j6 o# M& c  n9 ~2 F  iRelations:
4 t, B% p/ F: l( u9 P    5*$.1 = 0
7 X6 X" p% H- }
>> ClassGroup(Z) ;
             ^
7 I/ z- k# j0 L5 f, z: u+ eRuntime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

" E( u9 j6 @1 u9 LInteger Ring
Abelian Group of order 1

lilianjie

Z:=IntegerRing(12) ;Z;   
+ |% @& ]' t' z7 N9 t; } UnitGroup(Z);
MultiplicativeGroup(Z);
9 }) B9 m2 G$ c2 r* ACategory(Z) ;
$ e- A7 w8 k: U' S3 R  oPrimeRing(Z);
) q2 I- B: L9 G' KAdditiveGroup(Z) ;
+ `! h8 G# w/ ^. f9 S/ @6 S
Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
$ j6 c1 a$ R' B3 ]9 UMultiplicativeGroup(Z);
! b. U  l) W+ B% Y. ICategory(Z) ;
, Y7 g' G6 i" a& jPrimeRing(Z);
$ f  V: a- [8 f* i3 f9 J( }AdditiveGroup(Z) ;


+ P3 L/ n8 b) w$ h
Residue class ring of integers modulo 12
6 n, j. ^6 |" q- @/ F5 }Abelian Group isomorphic to Z/2 + Z/2
; L% S! _7 c- @2 b3 _) ^Defined on 2 generators
Relations:
# o& F! z- m) t8 ~- b    2*$.1 = 0
) @! [/ v: G6 o6 ?- [9 V+ m$ Y: m    2*$.2 = 0
0 y. N+ t4 \* v8 g8 h! k( bAbelian Group isomorphic to Z/2 + Z/2非素数环的乘群同构两个小群的直积（1*11    5*7）Defined on 2 generators
Relations:    2*$.1 = 0
9 j; d0 Y, W/ q/ w; S. O    2*$.2 = 0
6 `7 b& n* d1 Y) z  F  gRngIntRes
6 }% g) O6 S% D5 h1 o( h% K7 w7 `Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
, L9 J- t9 o& _. E+ f. r5 \+ DRelations:
* T: @) k$ P& x& ~8 p- E' F    12*$.1 = 0
6 I% @: ?1 p7 {2 b" L3 wResidue class ring of integers modulo 13
/ L. O- |  ?, |0 \3 R! }) tAbelian Group isomorphic to Z/12
Defined on 1 generator
/ |& `$ L- U, W1 M! M8 FRelations:
# W% l9 T6 R, f  N) R7 t    12*$.1 = 0
Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
. V, b/ ~, C' q" `; o    12*$.1 = 0
; }3 L$ w9 j7 Y3 ARngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
; m8 s7 V- B3 I/ p0 `3 l0 P3 Y. dDefined on 1 generator
! u9 z& U6 X) A: z+ N% \, }% d7 sRelations:
    13*$.1 = 0

lilianjie

Z:=IntegerRing() ;Z;   

R:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   


PrimeRing(R) ;
$ ?$ ~5 g. A* A+ s0 jCentre(R) ;
* N  x0 Z! \' k' L4 X
Characteristic(R) ;
- p9 h7 Q! R, A9 Q+ Y7 Q4 j# 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) ;
  Z3 g0 f* ?' n
5 a0 m$ ?8 \8 N* X3 g1 `) e% E  iIsPID(S) ;
IsDomain(S) ;
Has**(S) ;
R eq S ;
R ne S ;

Parent(R!123) arent(S!123) ;
4 J/ z+ a' Z9 E# ]9 v9 a8 _' m' PCategory(R!234) ;Category(S!234) ;
% }, Z* q* X3 t1 z: k! _
* U$ g  h3 U' U% R6 {# N. m4 Fa:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
7 ~2 Z1 G; W+ FRepresentative(S) ;

(R!a) in R ;
(S!b) notin S ;
- |5 a6 R7 E' K6 \) l8 x5 Q( iIsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
& [3 {" k  p5 I- }1 W6 y8 k9 @IsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
# K. b" o7 m4 ^IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
. x# z. y. `& K: a5 ~  [
) k, D! ?- B+ J: Q" ?% dZ!a gt Z!b ;
Z!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) ;
Minimum(Z) ;

Maximum(S) ;
7 g8 K, A4 W/ E4 j( uMinimum(Z!a, Z!b) ;
Minimum(R) ;


$ _1 {6 c! {# E3 _/ u5 o
! q0 V$ {( X& f! [( |, Y) }- lInteger Ring
0 Q: g4 p- m5 x/ e$ eResidue class ring of integers modulo 12
, @. l/ G" @6 _Residue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
12
12
false
# E* k2 j6 E/ Ffalse
true
true
true
; F7 N8 I$ J" V& m: Y/ }6 s3 x/ Htrue
false
) X3 h) K, _4 ]2 Ftrue
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
) a: f3 I! ^2 i* S. ]  I( x& \RngIntResElt
RngIntResElt
9
  j$ q% @7 k9 n( w12
0
, Y# q. W$ ]& I' ^% W% e0
' o5 C% O3 n! D! X+ U3 e  ]6 z( Ttrue
6 K5 t- E* x; xfalse
' [8 S* D3 U. A4 ~/ D/ p0 ]1 Ufalse
. l$ U( x9 P7 g! R+ Ptrue
4 N9 Z" c. [: q0 ^1 xfalse
true
( A1 j* z/ q' J' M  Ffalse
false
false
false
true
) D( V. Q# F( \0 ?# Otrue
12
1
# _, @1 J' d3 y: G/ \! u
>> Maximum(S) ;
          ^
: R$ Z3 W6 W* t0 }. _2 K# }Runtime error in 'Maximum': Bad argument types
/ I* S" }9 S( `* C9 iArgument types given: RngIntRes
5 ^4 F1 J% g7 X" H3 ]9 X
/ L0 y. y% j( o( J: `9

. ?* ?; x/ {+ M, o/ [1 Y1 t7 ^>> Minimum(R) ;
          ^
Runtime error in 'Minimum': Bad argument types
/ W9 Z9 g% U/ h* PArgument types given: RngIntRes

lilianjie1

8 O: B  b. [) {9 ~- J
- g- x! `2 R3 ]' j6 \+ l6 \! qZ:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
ZZ:=IntegerRing(15) ;ZZ;   
) u3 t, g  V! J, }8 g/ PIZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
ZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
IZZZ5;

I12 *  IZZ15;            理想和/积/并/交，
4 z6 Q# s8 c( g0 w理想和是理想对应两（可多个）元素加，
$ r; F" h5 K; _4 _7 w- E理想积是两理想（可多个）对应元素积，
8 u/ @* Z  h0 k9 d. h* n理想并就两（可多个）理想元素并，就不一定还是理想，
9 Y7 w; [: k& u. \8 L4 c. t理想交是理想（可多个）元素交，理想交一定还是理想，

理想积是理想交的真子集，极大理想交是理想------J根
/ Y  g4 J+ R6 N理想商就理想间同态：是必须能整除
5 J" t0 K, o5 I( yI12 +  IZZ15;
I12 meet  IZZ15;
8 v1 Z# x% N: h: i
3 d( ^" Q9 `! ~) F6 ~: vI12 * IZZZ5;
. U6 x! Z) {- i5 t* d5 B7 w0 `I12 + IZZZ5;
: |! K8 E6 n5 N0 b* a; _) @* H, DI12 meet IZZZ5;
' O% B# x  w! p% X( \I12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
' G% q. j7 o. x! NI12 + IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
4 R9 k6 d1 H# A: FI12 meet IZZZ5;
IZZ15 meet IZZZ5;
7 w% j7 e" w: G1 `/ _7 C! \9 |IZZ15 / IZZZ5;
8 `$ E# ]- \2 {* y4 v
; a3 I! \4 L5 q. N" ~0 F; W# lI12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
& @; K2 x, h, T$ N; YIZZ15 subset IZZZ5;
6 U: e2 b/ O' ]8 h! @9 m2 @IZZZ5 subset IZZ15;
9 K6 }) }7 E" r- s) u5 E( w  XInteger Ring
; K3 j' M# e) ~1 B) j, D0 WIdeal of Integer Ring generated by 12
& B  z% \; R  @7 o1 HResidue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
% m8 u7 l% H; |8 g& w3 _. f& t+ l; H# Hfalse
Residue class ring of integers modulo 12
Residue class ring of integers modulo 15
5 d! _: h/ {* e9 N- aIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
/ g1 M% {8 ]6 C' ~' D# J4 JIdeal of Integer Ring generated by 3
" l; f5 Q$ f- X/ A6 m/ ZIdeal of Integer Ring generated by 60
$ q0 c7 s7 V) e& v: N1 m$ ^Ideal of Integer Ring generated by 60
Integer Ring
Ideal of Integer Ring generated by 60
2 L/ x  F' k) ?6 z5 v7 I
>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.
: f, @( w9 i4 R' d( n) h
6 z6 f& F7 @# n6 d: M; x
0 T$ E. v% Q4 g1 G+ i9 k1 \>> IZZZ5/ I12 ;
( N3 O& E4 S/ m        ^
  }  {: a9 ?& Q" t2 VRuntime error in '/': Argument 2 must divide argument 1.
% w$ G' d* s( x( M7 U6 j/ X6 P5 Z9 y
. @0 m' d- K+ P  t( r) ?& x0 DIdeal of Integer Ring generated by 5
$ [7 C& Z$ f9 L- g; j# s  VInteger Ring
Ideal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
) Z5 e, T. ^2 K' l7 a1 hMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
4 o' A% _/ j2 r* K1 {, T& S" DIdeal of Integer Ring generated by 5
3 {8 P+ Q0 Z1 E8 `. {' }+ ^Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
" s' S8 w& |- O! N/ e0 @Ideal of Integer Ring generated by 3
' E) x+ V- `! R3 h* w% l5 t6 hMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
  b& l: \7 |. }4 A- F6 ~; x8 d( D
- j1 s* d! r! u* u; Kfalse
true
  h0 e+ P4 [# f2 q( i/ rtrue
7 T! n3 h+ Q2 cfalse

lilianjie1

Z:=IntegerRing() ;Z;   
+ Y9 e7 C2 t% vI12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
% K3 c( P4 O5 G/ K" x1 Z3 w, VIZZ15;
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> ;
6 F. b; E; t; ]7 {9 O0 Hloc< Z | 17> ;
loc< Z | 131> ;局部化：一个素理想到原环元素的映射
3 S0 l) c+ Z9 p. Sext< Z | > ;超越扩张到一元多项式
ext< ZZ | > ;

3 t2 z; V" K, }: u7 Y0 kext< Z, 2 | > ;超越扩张到多元多项式

ext< Z, 3 | > Completion(Z, I12) ;
& \* N  Q  e( ]' ]

. _6 {+ Y  P- \) c% c% q' jcomp<Z |I12  >;
8 s2 q# `) x9 P, Y    素理想零理想完备化，和P进环联系起来
2 T$ a2 y) g0 g" t' m: ]* vCompletion(Z, 0) ;
3 E! m5 Z  d. B( q4 i# Ucomp<Z |0  >;

% q# m0 K3 A" g) ~* T1 [5 [4 TInteger Ring
$ e* f/ x6 F- g, o) ?* }- kIdeal of Integer Ring generated by 13
! v$ U% X1 a$ G' y/ H" _Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
Finite field of size 13
/ s( k, ^9 l( N4 X9 l) yMapping from: RngInt: Z to GF(13)
modulo 13

* n$ u* r% A) i5 V1 _2 @, w>> ResidueClassField(IZZ15);
                    ^
3 x3 u  ?/ z) }4 ^/ k) H. Y3 V5 bRuntime error in 'ResidueClassField': Bad argument types
Argument types given: RngIntRes
1 m5 }- p$ D4 x4 X
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
  r$ A" r  B( x3 `Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
& T* [9 k0 D# Q' w5 B* S+ p) L7 D1 bMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
  y& \; O4 N/ p! fUnivariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)
' ^4 c4 \! @9 ~
>> ext< Z, 2 | > ;
9 a& a  \4 z' p8 B% t/ V/ t      ^
Runtime error: This constructer is no longer supported
) }- X) k6 w( J: j3 v# T. Z9 w
! `  ]9 k' d7 \1 J6 E
, Z' {' r0 _% y>> ext< Z, 3 | >
      ^
; u+ N& g0 {9 tRuntime error: This constructer is no longer supported

13-adic ring
$ W$ u9 d( H  [7 D( WMapping from: RngInt: Z to pAdicRing(13)

Completion(
/ F3 O% [1 g( S7 ~6 @    Z: Integer Ring,
* v+ x. s6 @: V, R3 x( G    P: Ideal of Integer Ring generated by 0