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;
8 N# _% b; F3 en := -1666666666234567890;
5 Q3 x6 J! T+ Z6 r1 ]> n;

> n:Hex;                           转16
6 u1 j0 M3 D- L2 SIntegerToString(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);
Identity(Z);                  
7 K) x7 D( [7 T& E4 ]Representative(Z);         环代表元
7 [6 g6 r7 D; pEltseq(n);                        取整
: J, {6 r' N2 q( G; oEltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);
. W0 U. S& M- J" x( J: M6 S3 Z
9 L0 g+ c( B( K# i' Z6 s: o* ]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>;
4 X* c, T" D% B% g* F3 h7 v> k;
n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
9 @' o# R! ^/ f: R9 X> kk;
$ v( X5 d' {9 ^1 E0 Q2 Y; \  Fkk eq k;

2 }/ x. ^* b  o! Yk := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
# T) U6 K: D2 \* L) l6 Ikk eq k;
' x" U4 G  o- Z; A
6 W, P- |, q! _, f  FEltseq(kk) ;Eltseq(-1/14);




k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
  n( |- j9 S. H> k;
9 M7 \( A2 l2 Wn eq k;
  ]  u5 C# E5 xkk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

: L& o$ v* B* r! z& nk := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
n eq k;
+ q/ E% T# K5 B/ ekk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
* e8 z6 N. G/ [) Y  @2 }2 |% x3 l2 `> kk;
/ G2 o; T0 i, f* Y  S0 A( _kk eq k;
+ j0 B  T: z8 W5 H7 A
+ q1 ^$ X0 l0 N8 R& c) B5 r% bEltseq(kk) ;Eltseq(-1/14);

  Q4 _# T0 B& m





3 P2 J8 r& e- D
& |1 E! |4 {$ K+ f, w

=============
; I: ]+ P3 i9 l$ Z


, Q) |/ m5 L: E0 t5 n
' G, B9 h% n+ e: D( f# s4 tInteger Ring
-1666666666234567890
-0x17213080A7E55CD2
3 a0 H% o& C* k1 J-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
% }, b4 G( v3 v-1666666666234567890
9 @: f0 z2 e; s0
1
0
! z, t5 V9 F, N- s[ -1666666666234567890 ]
2 f* U; D8 S, A5 q- X+ X' f! s- n[ -1666666666234567890 ]
1
13
1
2 h3 Y/ S8 K& v8 U
-1666666666234567890
true
-1666666666234567890
2 p4 {0 F6 i( _8 V& X: o( Ttrue
; {. t5 ^# L' _/ n( V" B( O-1666666666234567890
  P& H2 r- i+ q! j7 |" t1 d$ f  Strue
-1666666666234567890
. \9 [( k' C/ f! Jtrue
% e: a0 k. U, s1 r[ -1666666666234567890 ]
& U) Y/ t1 b* e/ F* ]8 ?/ n7 P. z1 k[ -1/14 ]
. S" g/ i. s2 V5 q0 R
5 \# F8 m0 @. k1 a" g

" R4 @( x; M2 r9 P
0 q2 x- L0 [! l. Y


-1666666666234567890
' r) ^1 l2 X' }, G-1666666666234567890
true
8 b, k! U& k4 @-1666666666234567890
true
-1666666666234567890
+ A: l* z: R2 F, ctrue
-1666666666234567890
+ s6 a1 [3 W7 [0 \# P& qtrue
[ -1666666666234567890 ]
3 a+ g' W+ \$ N3 W* k[ -1/14 ]

) y7 U2 l/ u. b# A; B& v

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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;
s:=0x12345678111;ss;
% E: U+ Z" q7 [6 M) ^* t) S+ g3 y" W
; T& p- {7 F0 o" l% b2 q! fsss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
" }& i1 D( ~" J& L1 HFactorizationToInteger(sss);
8 e8 [! \3 [8 e  P" p+ P) rFactorisationToInteger(sss1) ;
+ Q# S! q0 Y  n" GFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
5 `  W9 B, J. l% L- TSequenceToInteger(ssss, 2);
, @# Q: g5 i! O! x3 y" ?% vssss:=Intseq(ss, 17);ssss;
$ P. |( S4 V( w; K: q0 t) t" T2 P% ?! wSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
" V0 V  t; d( H) gSequenceToInteger(ssss1, 17);转成2和17进制

12345678111
12345678111
# P, ?1 X1 E$ n% A3 d1 t[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
" \4 q/ c7 R8 ~[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
0 J& K6 l& b  k1250999894289
[ 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,
+ L6 f  o+ P2 R1, 1, 1, 0, 1, 1, 0, 1 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
7 Y# V1 v) Q/ w  p7 m% ][ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
1250999894289

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作者: lilianjie    时间: 2012-1-11 13:12


( j* i5 G0 w/ @' S' b& F! e+ e# IZ:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
7 L8 @, g) ~2 ~% m4 w$ z( Gn1:=Z!1111111111111111111111;n1;
( [# {5 Z. H3 U% O" ^n2:=Z!11333331111111111111111;n2;
" m7 {, U8 D+ L0 a5 B3 x
) Q: |8 d% k$ Y! \9 j1 m/ Y$ n4 k9 @2 ^
K:=Z!n1+Z!n2;K;
0 y$ k# E2 l4 U7 h. M
: p+ j4 S1 d3 j% ~' i* OIsField(Z);      是域吗Characteristic(Z);环特征
" b7 f8 S1 h  _  R, w, z: xIsFinite(Z);有限环吗
6 J( t! _4 U$ t! U2 qIsCommutative(Z);可换吗
6 D2 ~5 I4 U% {9 X" PIsOrdered(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
7 ^" F4 D. _! P
IsUFD(Z) ;唯一分解吗
" L8 J/ m$ t- i5 H  Z( xIsDivisionRing(Z) ;除环吗
IsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
1 e* B2 w4 Y9 u2 V, I! qIsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
; P* f/ t: K+ f1 V" h, g4 vUnitGroup(Z);单位群
/ V# |8 o6 r- t$ IMultiplicativeGroup(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
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
7 P6 g: Y* M  s5 ~! F# t2 F5 c& `
ZZ:=IntegerRing() ;ZZ;
ClassGroup(ZZ) ;
1 o% Z% m2 F. u& ]5 e. A& z
===========
; E. c6 ]/ V  x3 P
Residue class ring of integers modulo 5
9 L: a% z4 P0 x, j0 K+ l1 j5 w1666666666234567890
1
0 C, C/ `# ]& |& C1
" k. N1 i. w) O+ D2
- e# H' U7 }  t. strue
  P6 _/ e+ S. ]  O, l4 b0 v4 y: H4 A5
! @- {2 x$ {5 o5 |+ Strue 5
true
* V7 a' w8 \. d1 kfalse
3 a0 Y6 @1 x( v% x5 o$ P5 A! Rtrue
% i3 t- b8 g( f# W! x$ e/ v+ ztrue
true
true
' N2 H" w( ]8 A, A( k) htrue
true
3 f6 h. {& o' w7 l$ a/ U5 D9 wtrue
& S9 S6 P9 I: h# r  |Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
" _1 \! _/ a7 p8 qRelations:
    4*$.1 = 0
- ^5 H1 C; y; q6 p" V3 ZAbelian Group isomorphic to Z/4
Defined on 1 generator
  |2 f- N) t  t7 h9 TRelations:
/ \$ P, {% j( e- {    4*$.1 = 0
$ i1 F. O0 n* O2 c+ t! Q+ mRngIntRes
Power Structure of RngIntRes
1 R- U! n7 Z; BResidue class ring of integers modulo 5
Residue class ring of integers modulo 5
8 @4 @5 O- W+ V, _0 |% QAbelian Group isomorphic to Z/5
- h$ w( g: F  ^. kDefined on 1 generator
. L3 r4 C; c. b, @Relations:
    5*$.1 = 0
+ t2 `# E" U' k( P! V) e
/ _/ E9 J# D/ X/ z' |7 j, B5 k>> ClassGroup(Z) ;
             ^
Runtime error in 'ClassGroup': Bad argument types
- h: P8 l6 v, AArgument types given: RngIntRes
) r9 ^8 A4 ?. `; T& a- [: M
Integer Ring
4 }6 |3 Z+ H6 E# b# c. u! kAbelian 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) ;
* ?) ^3 E8 |7 p/ G2 g1 lPrimeRing(Z);
AdditiveGroup(Z) ;

5 L1 [, g7 P5 Z. a2 K) l9 N7 pZ:=IntegerRing(13) ;Z;   
UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
. c2 ]6 R" M- S1 H% w# H5 f, DAdditiveGroup(Z) ;


( X+ A! e9 ^4 s/ t% [1 l
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
    2*$.1 = 0
- F# u, d) M+ a0 W0 [" P3 ?    2*$.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*$.1 = 0
- w. n- X1 M# W  g& _5 n    2*$.2 = 0
, O7 Z" R: O& b) q! ~RngIntRes
" h* h5 R" R2 f! {Residue class ring of integers modulo 12
+ j: U2 x0 L% H- m1 r( yAbelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
* [9 ?5 Z6 @7 M4 w8 Y) t    12*$.1 = 0
% s& o  b2 d3 t5 P3 AResidue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
: Z0 d' T5 [0 q# q/ BDefined on 1 generator
3 `$ _7 E# s8 p% y9 V. S+ ZRelations:
. ~3 Z6 k  G6 x  v. f    12*$.1 = 0
6 }0 E, R2 e' g3 I, b$ jAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
: p1 l$ c1 j# ~" N6 B% A, gRelations:
    12*$.1 = 0
0 ?9 \! @7 L9 y) I  iRngIntRes
+ s$ h* F, |' f# k/ ]- i; GResidue class ring of integers modulo 13
  S, c8 O# C' BAbelian Group isomorphic to Z/13
Defined on 1 generator
Relations:
% J, F/ B1 w  ?2 j3 F7 z# K    13*$.1 = 0

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作者: lilianjie    时间: 2012-1-11 14:24

: m% u( Y& D! w* l( b" q5 Y
Z:=IntegerRing() ;Z;   
8 Z; V+ y: U' V; J% Q
0 [+ X% n2 C/ S4 ]! m/ cR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   
6 @! c( ~/ c( c* V2 A. E0 r5 \

PrimeRing(R) ;
Centre(R) ;

Characteristic(R) ;
# R ;阶----元素数
7 L+ C; `# i& {) S' e, G) OIsPID(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) ;
0 N! R# u; P0 N% \0 R
9 b) j& w) g( C. g/ G8 QIsPID(S) ;
IsDomain(S) ;
/ ?! `! }6 H1 |. LHas**(S) ;
R eq S ;
( m1 v7 t, ~! A& g% ^# jR ne S ;

: `) x! b. Z/ \Parent(R!123) arent(S!123) ;
0 k2 U% y* Y. N$ O0 qCategory(R!234) ;Category(S!234) ;

$ ^4 U: k. E1 q$ na:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
Representative(S) ;

(R!a) in R ;
; y( L9 h0 e# z4 G9 K: K7 ~(S!b) notin S ;
IsUnit(a) ;                是单位吗
% r9 o8 h8 o6 Z3 e- ~IsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
7 j, B( F0 e( y4 f3 [. `  _$ V# ^; f$ A: YIsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;

Z!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) ;
, K' x' S6 n) w; q8 NMinimum(Z) ;

: X( a$ f5 n- y) ^4 G% oMaximum(S) ;
  t7 a: c, S- \3 LMinimum(Z!a, Z!b) ;
Minimum(R) ;

, r9 T7 D1 u. B0 G; T
  ~, t% b' g+ ^/ @
) \& j( J9 R9 Q( O$ e" OInteger Ring
3 n7 v$ u7 g" bResidue class ring of integers modulo 12
9 f) i% s' D5 Z5 I. i! VResidue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
12
" k, U- v6 A, |6 M12
false
6 ^( e' h, _) h& F! ofalse
0 E9 R5 J8 K8 n. \$ h& utrue
+ X: B( R0 C0 o0 d, [7 \4 w3 D0 {true
( F# u* p# P3 j& Z$ c, gtrue
* O2 h: \; ^8 R4 `( ~+ E5 S( a) btrue
  _8 L; s9 O/ o+ H1 vfalse
true
Residue class ring of integers modulo 12
) x: N3 q! b, L* w6 F) O2 G0 `Residue class ring of integers modulo 13
. ^+ g5 V, f9 W6 n8 X3 L, L+ R1 a& dRngIntResElt
RngIntResElt
9
12
  ?+ t7 f* x( r& A8 _0
0
, \. C9 i$ n5 b$ [! N+ T. Jtrue
; n, d8 [1 h0 e. S$ zfalse
2 y8 R+ [! V$ w" A. o2 s) [9 ffalse
true
: [0 v1 h( ?8 |7 mfalse
$ E! u! r5 n; ?2 {true
false
false
7 L2 P% Q( }9 N% g6 f; u, |false
false
true
true
5 z. j( Q7 f2 [# p12
7 c" t; w9 }+ Q8 Z1 ?3 ?3 O& }. l# i1

>> Maximum(S) ;
          ^
  D4 O9 T  I9 o9 T+ V; v! _Runtime error in 'Maximum': Bad argument types
+ C6 l2 ?+ N! }2 jArgument types given: RngIntRes
8 e, o4 x- u- J/ C9 q7 w$ m
9
& N5 A# l* o, i6 E( d. W) W
>> Minimum(R) ;
          ^
Runtime error in 'Minimum': Bad argument types
9 }8 B$ X7 a' ^8 u3 ]3 P! m9 {Argument types given: RngIntRes

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作者: lilianjie1    时间: 2012-1-11 15:48

3 K7 V& P1 ?+ j5 L8 j, k) ~
5 }5 T- V/ l2 L  D' @/ _! eZ:=IntegerRing() ;Z;   
1 n: Z4 i$ M. }4 X2 ?. CI12:=ideal< Z | 12 >;
I12;
ZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
( K4 w/ a! Z: Y2 |3 d; kIZZ15;
) M& N/ D7 m" f7 T5 @I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
; n/ ]( ^# @- t1 b2 FZZZ:=IntegerRing(5) ;ZZ;   
" V: V! s! L  ]; q  \( b+ pIZZZ5:=ideal< Z | 5 >;
0 F, L8 G! Q. S+ N  k2 C- }1 QIZZZ5;

# [0 q/ g7 {$ R5 u( }5 iI12 *  IZZ15;            理想和/积/并/交，
理想和是理想对应两（可多个）元素加，
, e, J4 T' Q8 Y! a5 T0 j* Z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，
; a0 `: Q6 w/ l: O$ y+ |理想交是理想（可多个）元素交，理想交一定还是理想，
+ s) ]% C) Z. m/ j& t
7 x* g. M. f3 W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;
* ?7 V, v% s( U! _I12 meet  IZZ15;
5 A; i1 g/ ^4 j" X) D, f& t( _+ D
I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
9 x% Y/ c7 j, _) h  WI12 / IZZZ5;
5 r" G$ n% U- x2 W: z7 Z' {$ z8 wIZZZ5/ I12 ;
Z * IZZZ5;
I12 + IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
% f4 l) I+ ~9 A! k% cIZZ15 meet IZZZ5;
; K# ?1 Q, T& |0 l+ `IZZ15 / IZZZ5;
) J/ H- Q8 r+ M, z) g/ L  ?; f; T! b
# O  X3 z9 C9 H" u" j* ?* j6 d* jI12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
IZZ15 subset IZZZ5;
IZZZ5 subset IZZ15;
6 f$ o  V; d0 R5 W  i, gInteger Ring
/ s; T; h. v1 `Ideal of Integer Ring generated by 12
) q% u3 Q% A, qResidue class ring of integers modulo 15
! s6 H. y" m3 R7 v' K( J  w3 lIdeal of Integer Ring generated by 15
, ~! P1 Z" K& c  X% t5 u: L1 G8 n/ e7 F. `false
Residue class ring of integers modulo 12
Residue class ring of integers modulo 15
, i( O, c* [) s1 {" jIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
( w4 _$ n3 P& O/ _7 B8 }$ |3 lIdeal of Integer Ring generated by 3
  [, s9 }0 S+ UIdeal of Integer Ring generated by 60
5 x, |' C7 n. l" L  mIdeal of Integer Ring generated by 60
1 a" M& R) Z% k; l% E$ U8 q- T7 IInteger Ring
+ R* x# \( g( x$ g. Y" X- EIdeal of Integer Ring generated by 60

  u% E4 Q/ O8 N% T1 C>> I12 / IZZZ5;
       ^
* h' m5 q0 t- g' b( MRuntime error in '/': Argument 2 must divide argument 1.


>> IZZZ5/ I12 ;
' ^; @9 X9 a1 o, P3 C; a        ^
Runtime error in '/': Argument 2 must divide argument 1.
+ e. V1 ^# ?7 C* \1 f$ f$ Z0 w5 r
& P. u: n7 d' d9 Y8 e) X5 q8 BIdeal of Integer Ring generated by 5
Integer Ring
$ M$ e' F$ }" p) n0 U( RIdeal of Integer Ring generated by 15
Ideal of Integer Ring generated by 3
( A/ D" s; K* S5 NMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
Ideal of Integer Ring generated by 5
Ideal of Integer Ring generated by 60
+ ]" z4 e: O7 }5 wIdeal 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
+ K/ l/ u' _$ |$ a) _
( N1 r. b5 N6 N4 F# G9 i, T/ {false
3 J! c9 p( Z  Z: t4 Utrue
( Q+ v( @4 M  v5 y8 }true
false

---

作者: lilianjie1    时间: 2012-1-11 16:39
Z:=IntegerRing() ;Z;   
. K, G; F  [# b, J! _& I! V4 [I12:=ideal< Z | 13 >;
I12;
: i* m& E( N' z* U3 d3 X2 GZZ:=IntegerRing(60) ;ZZ;   
4 _% q1 D# a, r! GIZZ15:=ideal< ZZ | 31 >;
0 U2 f# L6 a; p/ c; p* O, g9 OIZZ15;
: F) t: M& ^; I, K$ Z1 A" cResidueClassField(I12);
2 j! e3 |8 z& e/ DResidueClassField(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> ;
1 e0 E& t7 s) `9 T+ V8 y$ Rloc< 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 | > ;
6 [/ Z: P; S& r' k( \
ext< Z, 2 | > ;超越扩张到多元多项式

ext< Z, 3 | > Completion(Z, I12) ;


comp<Z |I12  >;
3 V4 A5 w: h3 ^  b* E    素理想零理想完备化，和P进环联系起来
- `( s& ~, b7 F+ }" PCompletion(Z, 0) ;
  H* |2 T( U: m7 @+ k: ^5 j% Pcomp<Z |0  >;

Integer Ring
/ j: _% B% ]! cIdeal of Integer Ring generated by 13
8 G. m: ]4 h* b: TResidue class ring of integers modulo 60
Residue class ring of integers modulo 60
/ }' Q( o1 |  J" M3 p  zFinite field of size 13
, d7 Q. B1 `2 x6 s; F; l5 [Mapping from: RngInt: Z to GF(13)
modulo 13
9 J( a8 O/ @$ B0 ^2 N
>> ResidueClassField(IZZ15);
; a2 ]) R: ?% g2 R' Y                    ^
8 m* ]# `( Q' ~0 kRuntime error in 'ResidueClassField': Bad argument types
$ J/ Q% I) _" ]3 a7 P) ~Argument types given: RngIntRes
6 C; G4 ^; \4 S/ E% D4 j
* e# |1 D0 x; z1 F; DValuation ring of Rational Field with generator 19
  y0 W, V4 x0 J$ ^Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
Valuation ring of Rational Field with generator 17
( i9 R* u: p3 F9 V- }9 sMapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
- }0 Y, J" p9 W: F2 s* t9 lUnivariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)
" D+ H1 z  |$ q, F# c0 a
>> ext< Z, 2 | > ;
      ^
" g# S' y9 R1 d0 RRuntime error: This constructer is no longer supported
& X& B, y( G! @5 K# W$ q
; X! C" r5 U2 C
>> ext< Z, 3 | >
% L3 J0 E! N8 T) {# K' f- M      ^
" H1 v1 |; h  K2 Q4 b2 P, H6 URuntime error: This constructer is no longer supported
- \% s( N1 l1 u" D
13-adic ring
Mapping from: RngInt: Z to pAdicRing(13)
$ ?% e( ~% r3 n7 s1 d8 D* n! J. s
. Y  {" x5 L/ |, f7 {! B+ kCompletion(
- s" f& J$ i0 N    Z: Integer Ring,
. ^3 o2 u1 i1 n$ \1 g) T3 N    P: Ideal of Integer Ring generated by 0
$ G: x! O9 V; ~: N2 L% N( u+ M

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