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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;
3 M. n! x( p+ u& s1 `. ~n := -1666666666234567890;
. {5 E3 f  T9 O( X9 T$ \/ v> n;

> n:Hex;                           转16
4 K3 H4 [. m1 yIntegerToString(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);
) l4 }' P6 r7 O* g3 J# B# JIdentity(Z);                  
Representative(Z);         环代表元
Eltseq(n);                        取整
5 G8 ^# ~5 G- TEltseq(-0x17213080A7E55CD2) ;Denominator(n);Denominator(12/13);Denominator(222222/111);

& d* K5 h7 H1 T+ j8 b  k1 o; s1 am := elt< Z |  -0x17213080A7E55CD2>;m;         在虚实2次域中进制砖换不变
/ w" ^+ X0 k3 j# B' z, bk := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
4 q, ~' |4 E, c' w# Gn eq k;
2 {% J+ o8 {) i5 e7 mkk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
4 l% v$ X) v0 z& S
3 ^* I! r/ C3 Ck := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
- E8 t& D+ |. k- t" g, H7 i& Jn eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
! L. U+ v/ {& K. J' jkk eq k;
$ p0 j4 V/ E! |) b- n9 u( d7 p2 h% k
! x' y7 t* L% i7 SEltseq(kk) ;Eltseq(-1/14);


: f  g& P, ~0 a4 h* l
- m) }; h# k# X$ J' Y
4 t! u* d9 v# q. Kk := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
n eq k;
# M; F. g3 ?! M' t, O7 okk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
. j8 v. n7 u2 `$ `2 l' [n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;

' ]3 q3 {/ g1 KEltseq(kk) ;Eltseq(-1/14);

4 @3 A8 [; r! t$ D' |" B
, T) |( D& ]; B



/ {  d* e& `" F4 |! I, L

; }3 p( Y9 N% Z+ b! r. m! h! m

- t- Q. M* v/ e6 K- [" P=============

2 }( u  l+ u- i) r+ Q6 U

5 Q/ b6 v+ W: ]0 ~
* E7 d  t. z' J% UInteger Ring
-1666666666234567890
0 _& Q/ `4 }2 a) R0 y+ M+ I* X! j6 e-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
0 ]5 a8 U# o8 M" Z$ u-1666666666234567890
-17213080A7E55CD2
) t) J( U" m; M7 F" P0 J-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
2 |: T* H$ K: {4 w# C/ `3 W$ D# ?" b0
1
' ^( Q3 e2 o4 N: ]) p% e) p0
$ d& _0 g& J! n( n[ -1666666666234567890 ]
[ -1666666666234567890 ]
1
1 O- j# w& `$ ]( B- C9 |% m13
1
! K- X/ ]+ A3 f+ L/ M3 [) V& U
2 w" ^# Q: m9 B1 H-1666666666234567890
- P( ?( F  V5 B# k: R; ftrue
9 u& a# Z4 u; X/ N! S-1666666666234567890
6 Y# e/ _* ^* {# J1 s  Otrue
-1666666666234567890
true
-1666666666234567890
true
[ -1666666666234567890 ]
[ -1/14 ]

# A  y8 q0 x& P+ T6 @; L

+ w( Y! @  L: z& ?" ^



-1666666666234567890
1 `6 O' f! l" \4 |-1666666666234567890
: M9 W: ?, h4 `) L  K; [$ }+ {true
" q2 \* |& k) H8 B-1666666666234567890
4 R3 ^$ ~4 {9 vtrue
4 ]1 ]6 t9 T2 H# v) e4 }9 q2 A6 Z4 p-1666666666234567890
true
) ?: P& d" b9 f% Y-1666666666234567890
6 f% a1 D0 @* W/ z) Rtrue
8 t1 a1 }5 ?6 c% V2 `6 s[ -1666666666234567890 ]
[ -1/14 ]
, }/ H. [) U  \5 f: ^# S

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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;
; ]$ N  i( X" g1 a, ]
sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
; B* m& C# t) d4 Z1 c! bFactorizationToInteger(sss);
) t: }/ F$ m! r. xFactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
4 P$ k  _' m8 I6 cssss:=Intseq(ss, 17);ssss;
6 R/ q$ ]# I; k$ b+ y+ QSequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
; e3 @; K' f* x+ MSequenceToInteger(ssss1, 17);转成2和17进制
6 G6 `+ V3 L" n* v% j3 O. V
: f' s, V' R+ [" M% l. B7 }12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
* i/ h7 A4 S. K% y1 W7 B12345678111
! u" z( g* u, \8 w1250999894289
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,
+ @( j& ^2 ^. @* Q2 d/ Y! |1, 1, 1, 0, 1, 1, 0, 1 ]
4 g/ L8 A. J. l12345678111
[ 8, 6, 6, 4, 0, 8, 1, 13, 1 ]
* q# d4 c" R% ]# Q+ m5 d9 N* b12345678111
[ 14, 6, 7, 11, 9, 15, 11, 5, 9, 10 ]
' J2 ~% C2 R6 B: W" h  J1250999894289

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作者: lilianjie    时间: 2012-1-11 13:12

4 u' O( k4 c: P4 |
8 Q; |: u" M3 [" [6 u( d0 {( qZ:=IntegerRing(5) ;Z;           模5等价类环n := 1666666666234567890;
> n;
n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
5 T2 M' z% g& |  C
9 T) A: v! i; x" ^! z) ^. E1 M6 y# C
K:=Z!n1+Z!n2;K;

IsField(Z);      是域吗Characteristic(Z);环特征
IsFinite(Z);有限环吗
IsCommutative(Z);可换吗
/ v; g+ i( p3 M& j7 @2 V. dIsOrdered(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
9 f. s: T% s" B5 S
% ^$ n! J* Y2 `IsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
; n- P4 S' l/ Z3 q4 K! p5 kIsEuclideanRing(Z) ;欧环吗
IsPrincipalIdealRing(Z) ;主理想整环吗
IsDomain(Z) ;整环吗FieldOfFractions(Z);分式域
UnitGroup(Z);单位群
. i3 v8 |7 l' n( k$ 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
7 k  ]$ o. C5 n, ^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
3 Y' Q( h- T% O( n2 ^  r$ u
ZZ:=IntegerRing() ;ZZ;
* M7 v# t7 [) i" WClassGroup(ZZ) ;
5 F# u: P* M  e; @% _& K* p
; V5 M" O2 Z7 m; ^: i===========

Residue class ring of integers modulo 5
1666666666234567890
+ z7 N. x8 E4 g! p8 o1
' a- l9 }% M$ ~8 ~% y1
% ?; F' f/ X! r, O, [) J3 W2
! p1 U) X, h' H8 C8 Ttrue
5
true 5
; N+ W, T) d$ [8 Mtrue
false
true
+ z- r6 e3 l+ E* z, ]true
* z( n- f, d! N) Ztrue
true
; H  }3 C- w) z( j0 }true
true
true
5 x' P! w- ], C; v5 O/ L# KResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
' f- C' |& t) B+ o7 p6 U; m$ g    4*$.1 = 0
' O! @( E6 _: b) h; {Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
' [8 i' |2 l  A$ P( E, S( k+ I    4*$.1 = 0
RngIntRes
3 N/ o6 ^5 {8 j9 \1 N( W2 c, ~Power Structure of RngIntRes
6 }6 m( v5 W8 k  n. e; p, QResidue class ring of integers modulo 5
8 c% M. W4 n* E, c+ t1 zResidue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
& d0 P/ @" T& m. V, Y$ w3 X" e. c% vDefined on 1 generator
% s  r( n" ]; f8 _2 ?: p: i/ PRelations:
' `  [/ T; k5 ?  U    5*$.1 = 0
% T; }" i# D) m, I; |: J
>> ClassGroup(Z) ;
0 v; m% Q6 M; A             ^
$ I8 L* e9 Z7 u5 WRuntime error in 'ClassGroup': Bad argument types
0 u# U9 b$ P1 @" W, D: cArgument types given: RngIntRes

Integer 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);
MultiplicativeGroup(Z);
Category(Z) ;
1 N6 x; H  X! Q$ u% @$ PPrimeRing(Z);
; Q( _' Y# h, y# p! WAdditiveGroup(Z) ;

7 ^" x7 [4 Z  j7 u! A- m7 \Z:=IntegerRing(13) ;Z;   
UnitGroup(Z);
. b6 {- B- d0 B9 r% @MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
+ D2 S9 |$ u: _4 M1 }AdditiveGroup(Z) ;
' B, y0 F' o* t1 P


Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
& K6 a1 B2 P# J  C/ n; ^5 p- LRelations:
+ I7 L  ~- k2 \3 X8 Y  |    2*$.1 = 0
    2*$.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
# K+ X& x3 \# g1 {2 M' qRngIntRes
Residue class ring of integers modulo 12
$ p2 y+ i% S( i7 j+ n9 H6 J! x: TAbelian Group isomorphic to Z/12
Defined on 1 generator
1 m) E# A; d3 t: w  aRelations:
  _' a/ q1 n  S& B+ z5 B/ K    12*$.1 = 0
Residue class ring of integers modulo 13
: t" g' F3 L  s/ P7 B6 `9 sAbelian Group isomorphic to Z/12
' s( s* [; b9 }" U# x9 ODefined on 1 generator
Relations:
    12*$.1 = 0
* c# _8 G" l8 hAbelian Group isomorphic to Z/12     素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
) g, N' \2 W' y3 k6 C  U    12*$.1 = 0
+ a- U4 |% |$ n0 y, k( V+ ?RngIntRes
% t5 r! |3 R( b% \Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
, N- m& K; \8 u  [! f' CDefined on 1 generator
Relations:
    13*$.1 = 0

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作者: lilianjie    时间: 2012-1-11 14:24


Z:=IntegerRing() ;Z;   
3 d9 h( P8 P5 y; k( m! R2 _0 M
) s% U+ H: H* ^: CR:=IntegerRing(12) ;R;   
S:=IntegerRing(13) ;S;   
. b  N7 z6 z1 z3 U- r* a4 P

PrimeRing(R) ;
Centre(R) ;
) R. ]. r" S; K" _- I; C
8 ^  M! b( U% A3 N6 ]! L# \Characteristic(R) ;
# 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) ;
4 i- W+ h+ O; k1 j; xHas**(R) ;
1 {8 N. z0 E& ?- I  m9 A
IsPID(S) ;
& ~( y8 x1 H5 H+ S8 D4 j1 IIsDomain(S) ;
Has**(S) ;
R eq S ;
R ne S ;
8 x* H& u: ]" Q7 Z) ]1 g
Parent(R!123) arent(S!123) ;
# S- l6 |. R' p+ ]Category(R!234) ;Category(S!234) ;
) D5 h  u5 O" t0 @  Y0 k* k  `
9 r+ X% l' d2 |3 X! O+ sa:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
. G4 f. P0 `/ \$ dRepresentative(S) ;

3 u5 c7 \, o% s; U: n(R!a) in R ;
" X& j) n+ T1 N- B( i3 E(S!b) notin S ;
IsUnit(a) ;                是单位吗
IsIdempotent(a) ;是幂等元吗
: _/ C' f2 Q5 I" pIsNilpotent(b) ;是幂零元吗
  `" D+ [- X/ Z3 m/ l2 |& [) cIsZeroDivisor(a) ;可除零吗
IsIrreducible(Z!b) ; 可约吗IsPrime(Z!a) ;
0 \- g. f. H$ Z9 V, c
" J* w  Y3 m' V5 wZ!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) ;
( l/ F% q4 e+ K3 GMinimum(Z!a, Z!b) ;
Minimum(R) ;
4 R5 c3 k( p9 l4 S

) E4 N0 A: a* N8 U& \) l
4 |3 k4 R2 X  ?4 l2 aInteger Ring
Residue class ring of integers modulo 12
9 |# T% A  z; q) k0 PResidue class ring of integers modulo 13
. A# [/ {$ T( @, n$ j5 ]Residue class ring of integers modulo 12
& g( s& L) x/ E  M! F" F3 q& D+ AResidue class ring of integers modulo 12
12
12
false
false
true
true
true
true
false
$ D) [  G: G- v! q1 xtrue
+ w# t5 |9 c, Q2 s" a2 iResidue class ring of integers modulo 12
3 `$ d8 v" }9 fResidue class ring of integers modulo 13
RngIntResElt
RngIntResElt
9
12
0
0
true
false
false
7 u' S6 t* V% ?* Q2 dtrue
7 l7 P& x* b' A0 ~$ W/ Ifalse
0 M6 A' A( c; c- ~" Ktrue
, ]% z6 ^' ?2 `. lfalse
9 e7 L( G" p: H" Q4 qfalse
false
false
% t, d) m/ _0 i. F$ v2 S( ktrue
true
1 c5 T7 o; k8 ~( d7 G12
1
" C* ^4 O6 Y$ G/ J/ Z: ]
>> Maximum(S) ;
6 X) t/ c% n+ {* U$ z          ^
* T  L: q9 J7 B# |7 G; `Runtime error in 'Maximum': Bad argument types
- c$ d  n4 b$ F: ?Argument types given: RngIntRes
6 Y; B' k. U9 ^! g
9
9 Z8 o4 R( U) _$ E
>> Minimum(R) ;
; S4 X( _8 S% T1 w& E          ^
6 o* j! U; L+ ?' J7 a/ q( ^7 i8 j+ pRuntime error in 'Minimum': Bad argument types
. q- C* ?7 ^8 f1 L2 _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 >;
  V# q8 `( A0 k5 d9 t( J. qI12;
9 l3 K4 v6 n- A8 dZZ:=IntegerRing(15) ;ZZ;   
5 L  y. u, c1 G3 i9 dIZZ15:=ideal< Z | 15 >;
IZZ15;
I12 eq IZZ15;
Q1:=quo< Z | 12 >;Q1;
# e3 A% e; a+ j$ J) [2 V( ?ZZZ:=IntegerRing(5) ;ZZ;   
IZZZ5:=ideal< Z | 5 >;
* ]% c# d4 F' g; z! m+ B4 D% d9 _IZZZ5;

% C5 t. F6 g; u7 G  w) z; C) ~I12 *  IZZ15;            理想和/积/并/交，
理想和是理想对应两（可多个）元素加，
1 a) ^! g  \7 U! j3 Y* k2 Y6 \理想积是两理想（可多个）对应元素积，
. P/ M/ G! V) Q1 `- A( i理想并就两（可多个）理想元素并，就不一定还是理想，
" x1 S& [  v% C& K6 Z8 v理想交是理想（可多个）元素交，理想交一定还是理想，

0 K' W% W  I. c" Q! n) q理想积是理想交的真子集，极大理想交是理想------J根
* m% o5 x% V# e  e9 q理想商就理想间同态：是必须能整除
: F) P( G5 [9 j. q, k8 v  TI12 +  IZZ15;
0 C+ z: f2 a% C3 dI12 meet  IZZ15;
5 X( Y- C* C& X+ [3 r
I12 * IZZZ5;
I12 + IZZZ5;
! R, D! t- \0 N' y, Q( x" N% Q3 k) LI12 meet IZZZ5;
5 [9 R- J& L* oI12 / IZZZ5;
, U1 E. v" ^" q+ SIZZZ5/ I12 ;
Z * IZZZ5;
' ^# [: R- G0 U9 p  a, z1 T: PI12 + IZZZ5;
IZZ15 meet IZZZ5;
/ `3 v" z: a4 C( k0 o5 rIZZ15 / IZZZ5;Z meet IZZZ5;
) s$ F2 L: V, G5 QI12 meet IZZZ5;
IZZ15 meet IZZZ5;
$ _3 z! W7 O5 C1 `IZZ15 / IZZZ5;
! T$ d* {9 b8 v) g% z$ ]2 O
+ L- l* l7 k/ b+ }3 U- bI12  subset  IZZZ5;运算后的各种理想互相是否包含IZZ15  subset   IZZZ5;
* ~- l0 i( L& S+ jIZZ15 subset IZZZ5;
. j5 x# Z7 c5 A. xIZZZ5 subset IZZ15;
9 }; g/ C' L* D6 O. |Integer Ring
; K7 x+ t0 F1 n7 l3 zIdeal of Integer Ring generated by 12
Residue class ring of integers modulo 15
* i6 _5 H- S$ e( ?* [9 y1 i5 g" ~Ideal of Integer Ring generated by 15
false
2 S+ Z5 s9 S" n2 _, IResidue class ring of integers modulo 12
9 M4 T6 s% T. y0 k6 ^5 KResidue class ring of integers modulo 15
& O, r, h: S6 t/ W: g% e" JIdeal of Integer Ring generated by 5
Ideal of Integer Ring generated by 180
! ~7 [1 [/ ?0 x1 C) T: {; @Ideal of Integer Ring generated by 3
Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 60
* \+ @/ F# D! b4 G. aInteger Ring
) \" K) I* J# h2 g5 w8 hIdeal of Integer Ring generated by 60
8 p( V; Z& d' ~8 h
>> I12 / IZZZ5;
       ^
Runtime error in '/': Argument 2 must divide argument 1.
& N8 d0 o9 D7 v. t% L: e
! \0 p1 w( t8 s2 \4 N' L/ e' K( \
>> IZZZ5/ I12 ;
        ^
/ B6 B- v2 Q- p9 U6 {Runtime error in '/': Argument 2 must divide argument 1.

" ]1 s/ v& u1 P) OIdeal of Integer Ring generated by 5
Integer Ring
Ideal of Integer Ring generated by 15
( R( W6 ~* G7 ~5 p  L3 C3 i" [! \6 Q0 KIdeal of Integer Ring generated by 3
, f+ i5 q0 ]9 R  r+ d8 f/ gMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z
  N9 R8 b- h& uIdeal of Integer Ring generated by 5
% c* d- O! E1 t3 l" d/ _Ideal of Integer Ring generated by 60
Ideal of Integer Ring generated by 15
3 |, y6 c0 c! C. }) J/ w: x1 SIdeal of Integer Ring generated by 3
) A7 F$ D7 N6 s, n4 JMapping from: Ideal of Integer Ring generated by 3 to RngInt: Z

false
3 V+ T$ _( u  `' T! I# K3 _2 atrue
* R5 H% d: L; T* @, c7 [true
false

---

作者: lilianjie1    时间: 2012-1-11 16:39
Z:=IntegerRing() ;Z;   
  z( H, t8 x7 o4 B8 aI12:=ideal< Z | 13 >;
I12;
( o  n# s7 k" V, h5 a( f/ YZZ:=IntegerRing(60) ;ZZ;   
IZZ15:=ideal< ZZ | 31 >;
' {8 X5 T# N4 h8 z2 ?3 }2 \, WIZZ15;
, J. G' Q% L4 s- g* w$ mResidueClassField(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
3 l4 i! V/ U; n( eloc< Z | 19> ;
loc< Z | 17> ;
  B( K3 R$ B8 W2 ?) {% Hloc< Z | 131> ;局部化：一个素理想到原环元素的映射
4 R6 E  }; f" H& p6 u4 D! g5 X5 Vext< Z | > ;超越扩张到一元多项式
1 ~: G/ ], J/ \. {) Kext< ZZ | > ;

ext< Z, 2 | > ;超越扩张到多元多项式
0 T2 P$ x) l. X' u7 n1 R
8 C" f/ A8 K) U  Iext< Z, 3 | > Completion(Z, I12) ;
" M% C8 n$ o0 v8 h; T# B8 J0 e" C! t

5 o  E1 G; M% A1 F2 b" x- h' fcomp<Z |I12  >;
$ ]/ b1 O% t  h    素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
comp<Z |0  >;
" o& z; I' {6 D' i: {7 K" @# x
Integer Ring
Ideal of Integer Ring generated by 13
$ ~$ I0 c3 x/ O- V7 OResidue class ring of integers modulo 60
Residue class ring of integers modulo 60
+ q& e2 x& {+ P+ F0 |- n% UFinite field of size 13
Mapping from: RngInt: Z to GF(13)
modulo 13
! m7 s: m; @( i& r" X
>> ResidueClassField(IZZ15);
                    ^
Runtime error in 'ResidueClassField': Bad argument types
7 p2 ~8 ]! I+ h/ J- C6 gArgument types given: RngIntRes

Valuation ring of Rational Field with generator 19
1 B" s7 V& h, `+ ^0 M3 s2 X6 ~Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 19
& P5 ^  V% U0 p# G9 w" XValuation ring of Rational Field with generator 17
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 17
" t/ d" ~& l. @Valuation ring of Rational Field with generator 131
4 K, @$ X; N# [, c1 e+ K( ZMapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
) {3 s: C' e9 hUnivariate Polynomial Ring over Integer Ring
2 q% I7 g* a- D5 \7 VUnivariate Polynomial Ring over IntegerRing(60)
  W. X3 S) G( C* {
2 q* E0 O$ e# d4 g9 I7 _8 H7 g) E>> ext< Z, 2 | > ;
0 w0 Q! r: E& I9 e( s3 x( l6 s! \      ^
6 N+ k" ^# v' R0 w: C$ ^Runtime error: This constructer is no longer supported


% g' G3 S3 t: _) h/ t. z>> ext< Z, 3 | >
1 ^) Y: q7 q" Y2 p3 S      ^
1 ^4 ]2 E/ a& M8 T1 }Runtime error: This constructer is no longer supported
' `6 ]+ @) h( q! |8 M6 K) O
2 I: U0 a3 g! J0 C# z, P13-adic ring
! m) v3 ]1 y  S5 g% KMapping from: RngInt: Z to pAdicRing(13)

+ }  t/ g: W9 e( P1 sCompletion(
& T# d' a! ]/ p4 y- X1 T: v- ~2 i    Z: Integer Ring,
3 n3 v! F7 Z) \    P: Ideal of Integer Ring generated by 0
( |/ _, S* U6 l( C/ _

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