数学建模社区-数学中国

标题: 一些初等函数： [打印本页]

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作者: lilianjie    时间: 2012-1-11 12:30
标题: 一些初等函数：
:=IntegerRing() ;Z;
n := -1666666666234567890;
> n;

> n:Hex;                           转16
" k; L7 L* C" _6 q4 J( z9 GIntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
IntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-0x17213080A7E55CD2);转串Zero(Z);
/ I! Y: N6 b9 W0 ~% RIdentity(Z);                  
Representative(Z);         环代表元
Eltseq(n);                        取整
Eltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
7 g% s! e' K: }  w2 N' r! z! v1 g
9 K% |0 N6 l1 X' ym := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
3 }/ w2 _" e4 [3 z$ X8 d5 Ok := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
. t5 s. M+ A3 C' }. R2 z7 An eq k;
: J" i7 l+ Z' [/ ]/ \" h* skk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
9 ^& f4 y, E( F" _2 s& ]: X0 r> kk;
5 [: a7 b0 q- q2 E& ekk eq k;
- O4 l8 K& }( _& l# T; i: ?. N
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
8 a3 o7 p/ u3 [4 t> k;
6 u) s1 @- |4 I; A7 Gn eq k;
- V0 M8 ]5 C, C& b# [! l7 jkk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
: o& W4 t, f% O* L$ Z; H3 o> kk;
kk eq k;

Eltseq(kk) ;Eltseq(-1/14);


7 z" ]5 A+ O6 m- ~! x3 J8 i. s2 X; l
! E  F+ Q# T2 n7 d/ L: m( A* a
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
  y+ D4 b2 V) |9 Y/ k6 x6 wn eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
$ o' u4 D6 v2 T7 A, D: |kk eq k;

) F* x# b- ?" k* Z8 e1 Ok := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
* Z2 i; h7 I- _: A0 @4 w: |n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
$ t. D% N1 q; j+ P> kk;
# z* a2 P) R$ G( E# m$ q6 |1 w4 }kk eq k;

Eltseq(kk) ;Eltseq(-1/14);


& K4 d5 _- G6 n. s' Y


  F* |, x1 Q" G8 H; i! B' g' \1 T
; V' k7 U' l9 X1 V
  ~8 V( @( a; D0 E$ c% X4 F0 v


% M& \4 n8 Q, q: X4 A9 S8 s=============

2 A! A; V5 b8 N& t


! u. ?! h; [4 P9 k* pInteger Ring
-1666666666234567890
-0x17213080A7E55CD2
  E3 t" A; ^% X* Z) ~" m$ X( j-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
& j8 ]" ~7 v" x+ F" }# M-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
0
1
8 ~# @& j. V0 Z! r2 t  Y6 _8 _' {0
1 N$ i% K, H! P1 D8 d7 D) L5 g$ b- k  d[ -1666666666234567890 ]
3 s' V7 `: Z# V3 J- O[ -1666666666234567890 ]
' F2 L/ r: q) _- D1
# p4 r7 B5 t) ?2 w13
1
2 T3 g9 q# C/ Z7 E( U6 r
1 H+ L! ~5 x/ p; ^8 [2 a7 g-1666666666234567890
* @: F/ f7 |' z% R# K7 Ctrue
-1666666666234567890
true
2 x7 |: N0 \# `, ^: v( @-1666666666234567890
true
-1666666666234567890
$ R4 ^- Z5 ~  d) _& a2 m4 j1 D' Qtrue
[ -1666666666234567890 ]
[ -1/14 ]


* ]# {& b- j( e2 B8 S

* q4 T# ?$ p! l8 s! o2 E9 M+ z
3 E, }3 V; J  ]7 t
$ s8 l6 R' Q% `' v1 N2 J
0 Y* Y! Y. w1 K8 f-1666666666234567890
$ Z% R0 t5 Y  j" U-1666666666234567890
true
-1666666666234567890
4 C; X7 r8 @9 t5 R+ Z" dtrue
: E: _) b: L# y-1666666666234567890
/ p- W3 Y, p" z, B$ I8 p; Rtrue
$ `$ y/ ?6 f8 R$ l$ j-1666666666234567890
true
: o& k7 {( V# q- m[ -1666666666234567890 ]
' E5 T( i- X" O, W[ -1/14 ]
! O8 {3 c  ^9 C8 j+ u3 E' B" r9 g

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作者: 孤寂冷逍遥    时间: 2012-1-11 12:42

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作者: lilianjie    时间: 2012-1-11 12:49
ss:=12345678111;ss;
* N& d. }3 l/ |# x. E( Ys:=0x12345678111;ss;

# V/ i2 F8 f2 n/ z  _2 V: v! f" Rsss:=Factorization(ss);sss;
1 c9 x& k6 N# a: e5 g3 p# Tsss1:=Factorisation(s);sss1;
& B1 E% f  {9 l6 v! V2 ], x0 r" OFactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
. J' Q# b- v# k* p& C: C$ t$ @- S7 VFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
) d* [+ k$ R; d% ~" E  pSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
3 e9 c' G. M% m( [* X0 h9 o( b% a# lSequenceToInteger(ssss1, 17);转成2和17进制
! {1 C0 E( c6 @# a4 G0 ^- Z
12345678111
, T. |  S7 A% Z3 X6 `% b1 r12345678111
/ P9 M; o! u/ {* L[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
3 @6 p2 ~  h: ~6 I5 R; }4 F[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
# x+ j7 \3 c! O3 ^- Z" M( @1250999894289
* d: i' Y7 j4 ~* o: p8 Z1 m[ 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 ]
2 W5 o& [. m" t8 L" P12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
- }: [/ N9 M! a" L; r. H# c1 v- e" T6 |1250999894289

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作者: lilianjie    时间: 2012-1-11 13:12

2 [) T* A" F2 G9 W& A
Z:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
8 v: H; z: Q6 J$ Xn1:=Z!1111111111111111111111;n1;
8 v- b/ S/ }2 J3 s0 hn2:=Z!11333331111111111111111;n2;

: j$ X9 f0 c; e1 q3 q
8 \1 x% P9 Z4 D* O  z1 p/ k1 `K:=Z!n1+Z!n2;K;
+ f5 l. u+ J! n6 s
+ B0 d; f- s% e4 K# L" GIsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
' X0 `4 l2 E! r- ^: DIsCommutative(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。。。。
8 V7 y2 t& e% l# r! V; s5 W2 \8 jIsPID(Z) ;主理想整环吗

IsUFD(Z) ;唯一分解吗
* U/ w1 G; l3 T1 I+ QIsDivisionRing(Z) ;除环吗
* I- z- z; g1 {: a7 x- ]1 jIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
1 D8 |/ x0 u( t# pIsDomain(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
5 [, T' A- ?5 c. r: v6 c' _6 cPrimeRing(Z);素环单环和本原环不同Center(Z) ;中心
5 u' V+ _; u% z- W) G$ D) kAdditiveGroup(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
; T/ Q6 ?0 Q9 Q' c* _8 E
ZZ:=IntegerRing() ;ZZ;
# v4 \4 |  ^2 Q- X3 fClassGroup(ZZ) ;

===========
, s, A* F6 o9 F- Z
5 \3 ^9 B' D9 e  b+ q$ ?7 NResidue class ring of integers modulo 5
1666666666234567890
1
1
2
5 D7 Z- b' f( b0 s' F) }true
1 ~6 F; i5 K7 o5
) |  o3 W! A9 H* h, z0 N$ ^( o8 Ztrue 5
true
+ U" v9 g7 C% yfalse
true
( X! s, K# u* n8 Ctrue
( c! U$ J" b" ]  ~  Mtrue
+ z$ m! n% T7 k, R6 I$ Z4 ytrue
9 S$ M: k6 Y% F2 d# r" w" strue
8 b0 K" X; f9 P9 A& F" S- i& btrue
true
Residue class ring of integers modulo 5
) I; X+ |$ C1 u6 c' B) g; cAbelian Group isomorphic to Z/4
# S; u: A$ R* i  @  t7 L2 j0 H2 {/ BDefined on 1 generator
' \1 l) s+ M! W" }  B: w8 s7 wRelations:
% `& a3 p* L" B6 E, r" Y' N    4*$.1 = 0
# {/ N3 q% u9 h) S$ a, JAbelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
0 o8 C* `% |3 i0 l/ z    4*$.1 = 0
RngIntRes
& g0 E  D3 a. L: n& _) t0 NPower Structure of RngIntRes
) Q- L6 f& v% hResidue class ring of integers modulo 5
7 I$ q# C3 ?& Z* U0 r% r  C6 \, HResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
' h. r$ ?( Y4 H) F. E5 z- A    5*$.1 = 0

>> ClassGroup(Z) ;
* g$ h( n+ [, V) Y             ^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes

Integer Ring
" f7 R9 s, y5 r) W: g& ?6 C2 y8 oAbelian Group of order 1

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作者: lilianjie    时间: 2012-1-11 13:52
Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
- s+ R1 @1 E0 \( p5 k' zAdditiveGroup(Z) ;

Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
* M( W" a  ~. @* @MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
& [% R: M% d9 r  d/ Q

5 f; ~) L: ~4 K* X9 K
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
+ k' {2 F5 L4 iDefined on 2 generators
Relations:
) s* r+ y/ s7 @3 j" B    2*$.1 = 0
: Q: [* ^6 w* Q' }% y% E( p( Z7 H0 U    2*$.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*$.1 = 0
    2*$.2 = 0
RngIntRes
+ H, M% @  r& G8 gResidue class ring of integers modulo 12
* I1 S' D1 r2 C/ a7 G% ?' xAbelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
2 m+ a" N) E& U    12*$.1 = 0
Residue class ring of integers modulo 13
/ U3 [6 j3 F# _# J2 @% jAbelian Group isomorphic to Z/12
' ?- k8 `" K! ^8 F* [% Y' e; E) N) g+ MDefined on 1 generator
Relations:
    12*$.1 = 0
$ r8 X0 ]8 X8 mAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
& C! L( \: h& k  A2 e    12*$.1 = 0
: R- |5 f/ D2 ERngIntRes
6 H  ?" R  \, z: v, zResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
; \& b2 A4 m6 e) ?Defined on 1 generator
Relations:
    13*$.1 = 0

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作者: lilianjie    时间: 2012-1-11 14:24

+ |5 b6 V& ^$ n$ i# w
Z:=IntegerRing() ;Z;   
4 H3 P. ]7 C3 W9 E$ Q$ P4 X4 V+ @, z
* o, {7 `" c( W: FR:=IntegerRing(12) ;R;   
+ [: f' ~% U- X( U3 @9 j* eS:=IntegerRing(13) ;S;   


PrimeRing(R) ;
Centre(R) ;

Characteristic(R) ;
# R ;阶----元素数
4 y8 k9 A0 S7 j" q' u4 E: K6 GIsPID(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) ;
/ G, Y7 S7 J( S/ u/ C0 H3 \Has**(R) ;

1 E; g) H- l6 _$ }! ?  rIsPID(S) ;
IsDomain(S) ;
0 H/ G( y" y3 j/ v" nHas**(S) ;
R eq S ;
8 H$ r6 ^3 _* R' s) e' [0 E5 SR ne S ;

Parent(R!123) arent(S!123) ;
4 w; g5 y) O0 d, j9 b2 qCategory(R!234) ;Category(S!234) ;
& h2 C) J5 \; M9 y& L4 s. z4 V
a:=Random(R) ;a;b:=Random(S) ;b;
! I% {' G, ^0 zRepresentative(R) ;
Representative(S) ;
) w0 Y4 ]* B6 k+ }! I
(R!a) in R ;
+ W/ u: y* ?7 W9 b  x# |6 e, h1 z+ P3 d(S!b) notin S ;
IsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
% J; @, o' q% g* A6 F! AIsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
+ |* ^+ D2 `% ^) v2 M* [$ U4 |IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
* O7 y$ I3 p# z3 E! ?7 o
Z!a gt Z!b ;
$ u1 p' X  n3 U2 k5 W0 F; u  oZ!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) ;
1 N" q# A* [- R8 h) }$ x4 iMinimum(Z) ;

Maximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;


( }5 {6 ~  O! R( P
. y; ~8 Q3 N5 i. y, r5 h2 zInteger Ring
1 G9 q" v' N( t+ ~3 _Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
Residue class ring of integers modulo 12
2 H7 u/ t$ |) aResidue class ring of integers modulo 12
12
/ x6 u/ h+ b& g1 n; b12
# S8 I0 F# j7 j& m3 }# l& Vfalse
5 t$ P, t8 U% @4 `7 R( r" o5 B; Afalse
true
true
true
true
6 H0 N& ]( `/ f( xfalse
5 E& \9 |9 U3 xtrue
Residue class ring of integers modulo 12
: e8 \' q; |) e' |Residue class ring of integers modulo 13
RngIntResElt
& ?4 C/ h: E( ?RngIntResElt
9
12
% B2 ^+ s: b' K; @/ G0
0
true
false
false
true
1 I$ |* f# q6 Z$ Ufalse
true
false
$ {8 i. F- |  q- c1 u4 C, bfalse
4 m: V& v# n, p4 Hfalse
5 v3 v+ l' N$ F$ E6 efalse
, s: X% i. p# X8 Utrue
' n, b$ w, J$ [1 }- Ntrue
6 d- G- I. k  v9 Q" b12
  [5 [) O" b- G1
7 x! k: O* a( g5 d# |' B
" L7 R; W' l  G0 p>> Maximum(S) ;
! z1 _4 S; ^' y. Z# d/ d7 `          ^
Runtime error in 'Maximum': Bad argument types
$ N; V. I6 \, p- I" G! Y2 H$ A+ p# ]2 ~; @Argument types given: RngIntRes
. v& U. t# a6 q/ m  ]8 y
9

& d, o0 x2 Z2 f" v6 j5 l9 L- D>> Minimum(R) ;
          ^
Runtime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

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作者: lilianjie1    时间: 2012-1-11 15:48


Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
I12;
+ W) U; Q( @" |6 k( y7 [) H; {ZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
$ v0 c3 ~7 z" ]4 x3 P6 z0 oQ1:=quo< Z | 12 >;Q1;
ZZZ:=IntegerRing(5) ;ZZ;   
$ n3 g) L0 W, p7 B. ~, O- }IZZZ5:=ideal< Z | 5 >;
IZZZ5;

7 N- W, \4 O  o) C, _' a7 m6 Q2 {I12 *  IZZ15;            理想和/积/并/交，
7 z6 t5 s  K6 Q  E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，
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，
0 |1 C" s# D9 d0 N# o5 p
理想积是理想交的真子集，极大理想交是理想------J根
: l: Z+ b6 X6 u* _5 b$ p8 _5 X# _理想商就理想间同态：是必须能整除
- v, t9 z9 y. D8 {) t0 ^; eI12 +  IZZ15;
3 M4 }' b( S( G3 O7 S& yI12 meet  IZZ15;

* a* C& V3 X; B8 S. P; e9 X/ hI12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
I12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
/ w. C5 a, Q9 C" J0 d% o% D3 n7 mI12 + IZZZ5;
6 G5 Q0 c5 }: Q- T' T7 Z( J; _IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
- Z( ~1 V, ^5 N# H, FI12 meet IZZZ5;
IZZ15 meet IZZZ5;
- ^+ L/ s9 e! G1 F* e4 m; xIZZ15 / IZZZ5;
: h  n( B% @. Z" A8 m( q$ Y& W* O5 @5 Y
: N1 Z; f4 S4 W4 z" @/ b' Z6 _2 [I12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
0 @' Z% b: ~' h+ Y* C2 o* yIZZZ5 subset IZZ15;
Integer Ring
Ideal of Integer Ring generated by 12
" s  }9 Y- Q/ \Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
: O! v, U4 L! e6 x2 k8 X8 i; vfalse
+ i4 Q- ^# p. R( J3 p: S0 nResidue 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
2 u' G2 P! @8 a( J5 E/ [  MIdeal 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
% P# o% F4 }5 l  J% E
>> I12 / IZZZ5;
) g8 Y5 G: R  p5 V; y3 Q$ P* q       ^
Runtime error in '/': Argument 2 must divide argument 1.


>> IZZZ5/ I12 ;
+ [! c( K* ~/ R# o7 w        ^
Runtime error in '/': Argument 2 must divide argument 1.

Ideal of Integer Ring generated by 5
Integer Ring
% P0 t  X$ w" a: Y! t0 jIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
0 R* a6 H* x2 P+ HMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
4 F' Z" s8 R8 z. r$ O6 rIdeal of Integer Ring generated by 5
7 `* P% o6 V7 C' j* Q4 aIdeal of Integer Ring generated by 60
: @; S$ E! l4 g5 T) L* K3 IIdeal of Integer Ring generated by 15
' C) g0 g8 H8 d9 L' U5 PIdeal of Integer Ring generated by 3
Mapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
9 P* L) Z4 t4 S" l4 T. ?4 @! x1 F
& E; |5 V; y0 Z9 p4 N% Kfalse
true
* P& f- ^% `7 B  S% f- z6 qtrue
false

---

作者: lilianjie1    时间: 2012-1-11 16:39
Z:=IntegerRing() ;Z;   
: _/ c3 V+ P8 I+ KI12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
% o9 i* ]" R* R: `. H- H$ MIZZ15;
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
  Q) V  @. y# o1 m; Oext< Z | > ;超越扩张到一元多项式
ext< ZZ | > ;
/ x9 c. D! H  M' ]/ F( M4 B
# Y1 N! O$ R2 v' Q0 T. h5 j9 q% E- fext< Z, 2 | > ;超越扩张到多元多项式
0 E6 e& P$ G; r6 S' I# }. d% b
ext< Z, 3 | > Completion(Z, I12) ;


) M( C+ x8 C3 e3 k9 T7 Gcomp<Z |I12  >;
. o9 C3 {  x6 B3 A  u2 K6 j    素理想零理想完备化，和P进环联系起来
& ?/ X9 }7 y1 K2 X, gCompletion(Z, 0) ;
% g  a+ r) i$ |2 E- S- q( xcomp<Z |0  >;
2 L" b/ u& K! \2 w
& G( Z9 ]7 I$ WInteger Ring
Ideal of Integer Ring generated by 13
. u; }9 I5 j; G2 O5 a4 vResidue class ring of integers modulo 60
+ O2 j8 `  a+ S" T0 {1 d8 yResidue class ring of integers modulo 60
Finite field of size 13
+ B8 A/ P. _* O  }* lMapping from: RngInt: Z to GF(13)
5 D: ]7 u& q" v! dmodulo 13
' j1 u7 m1 |$ r2 v( o
>> ResidueClassField(IZZ15);
                    ^
! r2 G) _) ^  |0 Q. k, S5 s$ bRuntime error in 'ResidueClassField': Bad argument types
2 B3 `: d- M* ~! w9 [Argument types given: RngIntRes
  W# Y& ~3 V; s! F
7 S# g  k# G: J1 d; ^( jValuation ring of Rational Field with generator 19
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
8 a- O/ Y4 [1 p1 ]; G1 G0 @Valuation ring of Rational Field with generator 17
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
& C( R8 i  r" ?, i$ r5 I6 k: AValuation ring of Rational Field with generator 131
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
# X& M* O4 K. OUnivariate Polynomial Ring over Integer Ring
$ Z& j1 c1 o+ kUnivariate Polynomial Ring over IntegerRing(60)

# _" S0 v' t' y6 P5 _>> ext< Z, 2 | > ;
      ^
Runtime error: This constructer is no longer supported


>> ext< Z, 3 | >
      ^
) ~/ ?8 L- w. T% U& n' L/ BRuntime error: This constructer is no longer supported

13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)

Completion(
    Z: Integer Ring,
    P: Ideal of Integer Ring generated by 0

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