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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;

8 X; F; z# M! S5 E& Z# X: l> n:Hex;                           转16
! ?, E4 g$ I. Q9 V. n5 aIntegerToString(n, 2);        转2
IntegerToString(n, 10);       转10
4 }# l' o6 z# ?8 R7 c' iIntegerToString(n, 16);        转16
IntegerToString(n, 36);         转36IntegerToString(n) ;
IntegerToString(-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);

m := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
0 d4 i1 A+ L: M1 w( O4 J8 @0 [k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
- c* V$ Y; i5 {7 M/ }n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
, [$ _0 b/ I2 S( A& p1 [( C( z> kk;
! D3 W7 N& o  `8 s1 }" zkk eq k;

k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
+ V# B. @9 S' O( S. k> k;
, Z" D7 o, G6 M) @4 a- V& Mn eq k;
, d1 W8 [" t$ B# d6 Kkk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
- Y. q- Q) }: }: g> kk;
kk eq k;
; W! w# ~# |& I; U. r1 J
Eltseq(kk) ;Eltseq(-1/14);


/ ^: h: I! e8 a0 z6 z
' s3 K& ^# B: m9 K$ D( Y) O" ?
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
- w1 U( o  d& }/ F8 J6 q( u> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
% Q% H/ T5 Q. F: f0 _4 C
+ M8 ~+ Y9 @: D5 h' c: Zk := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
- U( d0 m. \: N* K) {> k;
8 M5 z- \' l9 }' D2 Q7 [3 fn eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
3 v0 q. f# E5 I+ \9 L5 J> kk;
3 J8 d' F/ T9 Z4 o5 r( ^3 D, a# }5 bkk eq k;
& a1 S6 f2 R) @8 e; A
Eltseq(kk) ;Eltseq(-1/14);
1 g1 D0 {3 P  D, R

7 u4 ]+ D" ?( o4 j/ J% i

, {: \4 {4 k5 B* V* t4 s
' I  j; Z, G6 g  W

* p8 I4 K* ?" P# b2 z, R


7 D8 _/ _, t" Q, Y, k( y& r8 h+ A" M6 `=============
* g! o$ f0 p$ ^
- n2 _; a8 p3 V/ g( Z# @( d

1 S# k: {* v. E; h4 X; G
Integer Ring
-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
% m2 R2 S( S; w5 G2 t-1666666666234567890
* I/ w# [" C9 H-17213080A7E55CD2
5 S/ `8 p7 b* `- W-CNUO0WGPY9CI
" a, S5 e; ?1 A8 h6 Q  G-1666666666234567890
-1666666666234567890
  Z  Q0 K) O" I) |  Y- e+ B" ^0
1
0
1 z9 c9 |- f$ c! M  Q[ -1666666666234567890 ]
, b6 g; ~$ z3 {+ V[ -1666666666234567890 ]
1
13
1

& U$ Y5 v9 k- B% W3 O5 ]+ ]' w-1666666666234567890
true
-1666666666234567890
! j# ]; w/ I2 D8 otrue
2 i# c5 U9 Z) O8 p5 s4 d+ m6 R3 I2 V-1666666666234567890
true
4 Z$ H; T0 |; `-1666666666234567890
true
[ -1666666666234567890 ]
  l1 {/ k% ?) p  ^% @[ -1/14 ]
1 d, x8 X$ o9 f( k" i
$ Q' B2 j, m' c2 P' I
! B) o* _- x& c6 b9 y* j- t" X5 _


' i1 b' Q0 s) @0 h6 ~' o
% E* y7 i0 |1 E) }: u, s  B" I/ @
-1666666666234567890
" U$ l- k6 T% Z) d8 @! x-1666666666234567890
true
-1666666666234567890
% R8 _1 [$ W# H- a2 c! o$ Htrue
! N8 G; b) k5 I' X+ X0 k5 Y-1666666666234567890
true
-1666666666234567890
true
6 e5 I' O, Y5 T' ?2 ]" t$ B8 v[ -1666666666234567890 ]
[ -1/14 ]

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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;
( B, L4 W( a! b8 d5 X
0 l0 k5 }4 `* j& C& ksss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
% c  F2 A, h" r$ s  c: S/ v2 E1 G& @FactorizationToInteger(sss);
3 o5 v9 d  r3 L6 @. s* ]! @FactorisationToInteger(sss1) ;
1 `, S( O- d: @7 i8 s  zFacint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
- J# ~' D+ R' D% Sssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
: N* z( D' F7 G9 X) NSequenceToInteger(ssss1, 17);转成2和17进制

12345678111
% r- ]: t; j7 Q3 B# a12345678111
# y$ [' \+ ]% V$ T. y[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
! K$ N3 K( B3 S3 o& H7 u12345678111
# S- e- l# b, k$ r' j, I1250999894289
1250999894289
[ 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 ]
12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
12345678111
[ 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

5 ]1 _6 V3 D/ Q& m
' W+ M, c$ M7 z# e& @! {7 a; VZ:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
! [2 O6 X  h8 c, X* j4 E) R5 h> n;
n1:=Z!1111111111111111111111;n1;
6 h% K+ T' o# Xn2:=Z!11333331111111111111111;n2;

) J: E8 E6 t% N, I: ^, z1 P
7 m3 Q- B+ @1 p& a4 I9 l2 I" k" eK:=Z!n1+Z!n2;K;

4 f* ?  b8 j2 NIsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
8 S+ T6 c# A! i1 [. tIsCommutative(Z);可换吗
$ R. W5 I  |9 ?. i5 l+ Y# m1 @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。。。。
: R2 L  X  z+ S% r6 _IsPID(Z) ;主理想整环吗
1 W- z% r  d: f5 k- E2 J
( E4 m8 D* z0 W; |( h( dIsUFD(Z) ;唯一分解吗
  M! l7 r' c* T1 M$ W8 AIsDivisionRing(Z) ;除环吗
9 J% `( {9 W- r; @5 w' yIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
8 ^  ?3 O/ N. YIsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
; b$ g9 O/ D8 q3 {UnitGroup(Z);单位群
+ ^' S; I9 E" Z, r3 DMultiplicativeGroup(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
2 n& I% C) o* E( b, t
8 W5 m# D( P% xZZ:=IntegerRing() ;ZZ;
0 m: t4 j0 M2 a# w; rClassGroup(ZZ) ;
/ g% ]; J6 e4 K6 j
, ]9 l- y8 s5 f0 d===========

Residue class ring of integers modulo 5
* h! R+ B- e: u5 R1666666666234567890
- J' [, \. @' w% m, w/ V$ Z1
+ U# R+ C1 d* b1 Q1
2
true
5
: V" ]- O1 B3 R( l$ W: L: y! Utrue 5
5 K' G' F# D2 j/ z( d8 S; L. xtrue
6 F" k+ M, E5 `9 k$ d  W' Tfalse
true
true
true
true
true
  G1 s$ f, }% J% N$ n1 f# U* Mtrue
1 _( [1 a6 }. o* M* L6 o4 U' x) Etrue
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
* N  U7 x9 y$ BDefined on 1 generator
+ b" @3 ^! H! ?% Z9 bRelations:
# @  n9 P2 a" x3 C    4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
    4*$.1 = 0
: I& D& p" p0 M$ QRngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
* m: S* f% E. M) f0 X- AResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
( [- }% U( s8 p; U; j- nDefined on 1 generator
& ^+ f2 t( X. B& ^3 F2 u* U3 PRelations:
    5*$.1 = 0
* P- n  g) s) G- B8 x+ Q
8 w6 O$ m6 z+ e>> ClassGroup(Z) ;
- [( u7 H1 c3 C             ^
! l! B; Q9 k3 ^) q% e6 D" `Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes
0 o1 O3 L$ v8 Z% ^; r; N
' r  F3 g  e, P( l; PInteger Ring
Abelian Group of order 1

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作者: lilianjie    时间: 2012-1-11 13:52
Z:=IntegerRing(12) ;Z;   
UnitGroup(Z);
- c" d, ?( X! |7 @MultiplicativeGroup(Z);
2 S# J0 y- y& ~3 Y* f$ QCategory(Z) ;
, Z, x' }' _; M- V9 zPrimeRing(Z);
AdditiveGroup(Z) ;

& ?* w% Z& p/ S# k" [Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
  ]( x# _" v+ j& b+ l; E" ~MultiplicativeGroup(Z);
. P% I# P% P8 G% `+ \3 VCategory(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
: f# m  y5 r, R, b$ l
8 v* C; y6 J) y4 g
* g8 g7 C: m  r, e0 I
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
0 u) H) I( n' C5 @. o: ]" n    2*$.1 = 0
; M; v* B+ C2 y8 c5 F8 A+ |9 d: N    2*$.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
" b6 L; ]% i' u- IRelations:    2*$.1 = 0
    2*$.2 = 0
6 J1 M# O( \, L! NRngIntRes
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
$ r- V5 D* F& RRelations:
6 s$ x8 }* q0 a: q    12*$.1 = 0
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
    12*$.1 = 0
# D1 u) k  R' u4 u0 i2 }Abelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
4 n+ {+ a3 x1 e% p0 u0 H$ H# iRelations:
  ~/ R3 G# v5 Z9 S1 J9 M+ r. M    12*$.1 = 0
RngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
Defined on 1 generator
Relations:
    13*$.1 = 0

---

作者: lilianjie    时间: 2012-1-11 14:24

$ O7 h( |: O& T$ h- {; E8 B2 K3 I
5 O3 R, j) n* H6 [, W0 `Z:=IntegerRing() ;Z;   
. g" T/ I/ j# V/ Z5 N; j
: X8 K" @0 Z8 H& X% k" sR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   


PrimeRing(R) ;
Centre(R) ;
4 r. ?$ e$ z0 L
Characteristic(R) ;
' x9 K, G, e) @/ {! q) c) 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) ;

IsPID(S) ;
IsDomain(S) ;
# f; I! [& z- n3 pHas**(S) ;
9 f1 M8 p  p3 f4 ?R eq S ;
R ne S ;
2 `& r- }$ J1 h6 A4 @0 a5 O6 t
7 ^) a4 b. ]5 a5 W" e7 CParent(R!123) arent(S!123) ;
/ ^+ n9 V" T0 ECategory(R!234) ;Category(S!234) ;
) V& I+ l8 `8 R; c* d
9 n2 ~; Z" v& ]* s. k' ma:=Random(R) ;a;b:=Random(S) ;b;
, M# V) w) ]9 ?6 h7 M9 LRepresentative(R) ;
' K/ X. z& ]! p4 Z  LRepresentative(S) ;
& R# |5 a* r& n+ B& ]
2 I& B6 a8 a: J0 \6 }(R!a) in R ;
  L$ n  |7 u! @9 D4 L& |(S!b) notin S ;
IsUnit(a) ;                是单位吗
: B9 i# f& z; L+ ^6 E$ @8 [IsIdempotent(a) ;是幂等元吗
4 Y7 @  w# K/ w6 a0 O" n0 l" cIsNilpotent(b) ;是幂零元吗
IsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
  e! c% h& \  I) j
& M, x: E2 o; ]5 |Z!a gt Z!b ;
# J7 s' u- ^6 F( A7 ~4 MZ!a ge Z!b ;
Z!a lt Z!b ;
) w# Y. K$ t) b+ MZ!a le Z!b ;只有同类环才可比较元素大小，Maximum(Z!a, Z!b) ;
) `/ O6 D/ A+ T0 B$ s" k7 VMinimum(Z) ;
% h. y) r, g, \- z
Maximum(S) ;
$ w9 u% [' ?/ B; }4 U# `Minimum(Z!a, Z!b) ;
8 O( {. @  l, }Minimum(R) ;
, L: {% _/ p$ G/ R" }' r
: _/ X6 U0 d' o) t' k( f

! d& k9 D. n1 Q7 B1 n1 ?0 OInteger Ring
1 Y/ x$ A+ U- E' f6 BResidue class ring of integers modulo 12
6 ~1 _/ K1 d9 }5 w4 f- o& LResidue class ring of integers modulo 13
Residue class ring of integers modulo 12
8 o3 D$ @4 d8 HResidue class ring of integers modulo 12
12
" J/ z' T9 q) T) j9 X12
false
# u9 F1 {( z  f. e: rfalse
true
# f2 i( c- ]/ S4 }true
true
, J6 i0 y* W* s" \; otrue
false
7 D" Y7 H" c. n, z% e' Utrue
- j! ~7 a3 t& c! O8 RResidue class ring of integers modulo 12
% _1 ?3 f4 ]: d: t8 S2 ~$ V* CResidue class ring of integers modulo 13
RngIntResElt
RngIntResElt
9
! Z* k7 n5 v7 K' y12
0
5 j- Q: W' x) J  O) \- z6 Q5 k6 P0
true
) v" G. S# S; r- T& zfalse
' b  J6 R+ Q, D6 x: f; kfalse
true
false
6 _3 P! L: a  d. O9 H" T3 Vtrue
& c' O6 D; m! l/ Q4 Ufalse
( N0 y$ [# i# S- W8 wfalse
false
false
0 ~! \7 z/ n9 H1 Atrue
true
0 w* G  P* T( C12
1

>> Maximum(S) ;
          ^
/ R% ^. O7 b. W4 I1 SRuntime error in 'Maximum': Bad argument types
Argument types given: RngIntRes
5 G8 o; z+ l& o6 `( m  }4 p! D( }
9

5 e5 k* W9 Y+ m; |' ~: B; F4 V>> Minimum(R) ;
/ |' i/ M6 N4 l2 \7 l, b+ s( Q          ^
Runtime error in 'Minimum': Bad argument types
Argument types given: RngIntRes

---

作者: lilianjie1    时间: 2012-1-11 15:48

, }2 K/ f' k, i" H$ M" C
Z:=IntegerRing() ;Z;   
I12:=ideal< Z | 12 >;
  \: B& K/ t; q9 W! Y3 y0 k+ S5 H4 @I12;
! P) o* L. h- f. @' lZZ:=IntegerRing(15) ;ZZ;   
IZZ15:=ideal< Z | 15 >;
$ }. c. q' @) ?  a; |, RIZZ15;
I12 eq IZZ15;
: L  p+ q" B0 \2 qQ1:=quo< Z | 12 >;Q1;
8 l, n' |7 C4 XZZZ:=IntegerRing(5) ;ZZ;   
8 J( x6 k1 J6 ~% eIZZZ5:=ideal< Z | 5 >;
9 K5 Z% d" O" F6 k+ G  Y9 K$ S1 I' FIZZZ5;
# ^. x( _2 v+ y/ `* j! Q' Z6 V
1 n! D8 I" y( p) hI12 *  IZZ15;            理想和/积/并/交，
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，
# R. t" s$ @( i, c: I6 {理想并就两（可多个）理想元素并，就不一定还是理想，
9 U$ x% F$ {$ [9 r) a& V1 S9 C* A理想交是理想（可多个）元素交，理想交一定还是理想，
1 t( Y+ J! r! z; {1 A' C3 ^3 Z; y
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
. @) A- q8 Q5 c9 rI12 +  IZZ15;
I12 meet  IZZ15;
1 p$ F5 ~% D6 j" n2 M# s: O
I12 * IZZZ5;
I12 + IZZZ5;
2 v. N8 I- e# h, G8 [5 S& KI12 meet IZZZ5;
I12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
0 _5 J7 k: O7 u. pI12 + IZZZ5;
4 t! ^( [) u$ D6 b/ {& eIZZ15 meet IZZZ5;
9 v- G% k9 r8 c' ^7 VIZZ15 / IZZZ5;Z meet IZZZ5;
8 C3 Y$ ~0 V) a3 JI12 meet IZZZ5;
0 p2 a7 R' [5 z( ^+ pIZZ15 meet IZZZ5;
IZZ15 / IZZZ5;
; {+ }3 ?6 s2 W5 h7 F
: j3 h( Y- l- S3 ~I12  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;
3 S* |& ?" W( J$ a- wInteger Ring
0 m# R, x" f  W) Z! |Ideal of Integer Ring generated by 12
: x' c. i/ l2 i/ x  NResidue class ring of integers modulo 15
# p! A9 K% Z7 ~1 h0 z( P6 RIdeal of Integer Ring generated by 15
; w2 w6 o0 ~8 j; f4 lfalse
4 m7 d) ]! j& u; d& \- U7 ?4 wResidue class ring of integers modulo 12
. x4 L, I" U; f, J, v0 t; v6 vResidue class ring of integers modulo 15
+ S" s3 D2 n. k% n% rIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
Ideal 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
; A2 z5 b, y- a" o& x
0 ]* h, K* e4 e* d>> I12 / IZZZ5;
% H2 I, u" A7 L: _! C/ f+ H. x       ^
Runtime error in '/': Argument 2 must divide argument 1.
- o6 A" q6 x* {( h% L9 ^
  F* z, U% c2 W& M( x/ H# C
6 _" ?' |5 {9 d>> IZZZ5/ I12 ;
8 o% ?( j: H2 Q0 F1 o. p! V        ^
  I% j# ]5 b6 [. D: zRuntime error in '/': Argument 2 must divide argument 1.
9 V, r  J+ j2 j4 x  c& ~
Ideal of Integer Ring generated by 5
" m. F/ y5 C1 Q4 c2 f% ^Integer Ring
Ideal 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
7 n5 u" h+ n) x0 X7 s8 qIdeal of Integer Ring generated by 5
& W9 G6 Q6 u$ \. R! PIdeal of Integer Ring generated by 60
: T4 w9 K0 L/ N" D  H& a: KIdeal 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

1 D( i' W5 @3 `, R' q% wfalse
true
true
& d, M& x+ }* `- pfalse

---

作者: lilianjie1    时间: 2012-1-11 16:39
Z:=IntegerRing() ;Z;   
% M2 z8 P" V5 R9 C$ c: _- ]I12:=ideal< Z | 13 >;
7 w7 |6 {+ @7 n/ i/ Q7 F) wI12;
) i' C$ p! _" M0 iZZ:=IntegerRing(60) ;ZZ;   
' Q' R0 M1 b3 S1 X& X& ~IZZ15:=ideal< ZZ | 31 >;
6 J6 H5 `  \  |IZZ15;
ResidueClassField(I12);
& n# T% d+ L7 m$ Z9 `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
  Z* j9 F6 \* r+ z  v1 Y* \  N7 rloc< Z | 19> ;
loc< Z | 17> ;
loc< Z | 131> ;局部化：一个素理想到原环元素的映射
1 W" Q$ s( Z6 next< Z | > ;超越扩张到一元多项式
ext< ZZ | > ;
7 l  c, ^& l: {) ]
0 P4 I5 O( \( F: R8 Gext< Z, 2 | > ;超越扩张到多元多项式

+ Z9 B  O$ e% @) e; Y: xext< Z, 3 | > Completion(Z, I12) ;
1 n) y0 v  p% M* a
. @3 m4 k% ^2 A4 `' n
comp<Z |I12  >;
; P6 A9 z" c) b5 }    素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
$ U( ]/ J1 U7 x2 M; B1 V2 [5 rcomp<Z |0  >;

$ F5 _( D7 y5 z, X' d7 ~Integer Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
2 D( N2 a# M2 E, ~6 l8 f3 TFinite field of size 13
  L8 E5 k3 f2 c. kMapping from: RngInt: Z to GF(13)
modulo 13

- B) `9 Y% h& b* k* s, ?0 M3 v>> ResidueClassField(IZZ15);
                    ^
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
8 W  C4 ^$ p5 u3 CValuation ring of Rational Field with generator 17
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
Valuation ring of Rational Field with generator 131
5 _0 u6 M# O2 I: p0 P* B' t! f8 WMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)
* z- y8 ^1 ~, p& d" q! O, v
>> ext< Z, 2 | > ;
$ ?4 P( N8 G, ?; e1 Z3 ]% D      ^
Runtime error: This constructer is no longer supported
. j  ?/ |+ }* V0 K+ a

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

13-adic ring
0 f3 v& C3 D9 Q  H, j# KMapping from: RngInt: Z to pAdicRing(13)

' M& T* [% @7 E; l% BCompletion(
5 O: n  n% X" g    Z: Integer Ring,
; v  Q- D; @! L5 E4 j    P: Ideal of Integer Ring generated by 0

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