一些初等函数：
:=IntegerRing() ;Z;n := -1666666666234567890;
> n;
> n:Hex; 转16
IntegerToString(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);
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
k := Z ! elt< QuadraticField(3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
k := Z ! elt< QuadraticField(13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
Eltseq(kk) ;Eltseq(-1/14);
k := Z ! elt< QuadraticField(-3) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-3) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
k := Z ! elt< QuadraticField(-13) | -1666666666234567890, 0>;
> k;
n eq k;
kk := Z ! elt< QuadraticField(-13) | -0x17213080A7E55CD2, 0>;
> kk;
kk eq k;
Eltseq(kk) ;Eltseq(-1/14);
=============
Integer Ring
-1666666666234567890
-0x17213080A7E55CD2
-1011100100001001100001000000010100111111001010101110011010010
-1666666666234567890
-17213080A7E55CD2
-CNUO0WGPY9CI
-1666666666234567890
-1666666666234567890
0
1
0
[ -1666666666234567890 ]
[ -1666666666234567890 ]
1
13
1
-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
true
[ -1666666666234567890 ]
[ -1/14 ]
-1666666666234567890
-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
true
-1666666666234567890
true
[ -1666666666234567890 ]
[ -1/14 ]
{:3_59:}{:3_59:} ss:=12345678111;ss;
s:=0x12345678111;ss;
sss:=Factorization(ss);sss;
sss1:=Factorisation(s);sss1;
FactorizationToInteger(sss);
FactorisationToInteger(sss1) ;
Facint(sss1);因子分解和还原ssss:=Intseq(ss, 2);ssss;
SequenceToInteger(ssss, 2);
ssss:=Intseq(ss, 17);ssss;
SequenceToInteger(ssss, 17);
ssss1:=Intseq(s, 17);ssss1;
SequenceToInteger(ssss1, 17);转成2和17进制
12345678111
12345678111
[ <3, 1>, <13, 1>, <31, 1>, <1447, 1>, <7057, 1> ]
[ <3, 1>, <83, 1>, <34129, 1>, <147209, 1> ]
12345678111
1250999894289
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 本帖最后由 lilianjie 于 2012-1-11 13:31 编辑
Z:=IntegerRing(5) ;Z; 模5等价类环n := 1666666666234567890;
> n;
n1:=Z!1111111111111111111111;n1;
n2:=Z!11333331111111111111111;n2;
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
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。。。。
IsPID(Z) ;主理想整环吗
IsUFD(Z) ;唯一分解吗
IsDivisionRing(Z) ;除环吗
IsEuclideanRing(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
MultiplicativeGroup(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
ZZ:=IntegerRing() ;ZZ;
ClassGroup(ZZ) ;
===========
Residue class ring of integers modulo 5
1666666666234567890
1
1
2
true
5
true 5
true
false
true
true
true
true
true
true
true
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0
Abelian Group isomorphic to Z/4
Defined on 1 generator
Relations:
4*$.1 = 0
RngIntRes
Power Structure of RngIntRes
Residue class ring of integers modulo 5
Residue class ring of integers modulo 5
Abelian Group isomorphic to Z/5
Defined on 1 generator
Relations:
5*$.1 = 0
>> ClassGroup(Z) ;
^
Runtime error in 'ClassGroup': Bad argument types
Argument types given: RngIntRes
Integer Ring
Abelian Group of order 1 Z:=IntegerRing(12) ;Z;
UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
Z:=IntegerRing(13) ;Z;
UnitGroup(Z);
MultiplicativeGroup(Z);
Category(Z) ;
PrimeRing(Z);
AdditiveGroup(Z) ;
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.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*$.2 = 0
RngIntRes
Residue class ring of integers modulo 12
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
12*$.1 = 0
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/12
Defined on 1 generator
Relations:
12*$.1 = 0
Abelian Group isomorphic to Z/12 素数环的乘群同构Z13-1=Z12Defined on 1 generator
Relations:
12*$.1 = 0
RngIntRes
Residue class ring of integers modulo 13
Abelian Group isomorphic to Z/13
Defined on 1 generator
Relations:
13*$.1 = 0 本帖最后由 lilianjie 于 2012-1-11 14:25 编辑
Z:=IntegerRing() ;Z;
R:=IntegerRing(12) ;R;
S:=IntegerRing(13) ;S;
PrimeRing(R) ;
Centre(R) ;
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) ;
Has**(R) ;
IsPID(S) ;
IsDomain(S) ;
Has**(S) ;
R eq S ;
R ne S ;
Parent(R!123) ;Parent(S!123) ;
Category(R!234) ;Category(S!234) ;
a:=Random(R) ;a;b:=Random(S) ;b;
Representative(R) ;
Representative(S) ;
(R!a) in R ;
(S!b) notin S ;
IsUnit(a) ; 是单位吗
IsIdempotent(a) ;是幂等元吗
IsNilpotent(b) ;是幂零元吗
IsZeroDivisor(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) ;
Minimum(Z) ;
Maximum(S) ;
Minimum(Z!a, Z!b) ;
Minimum(R) ;
Integer Ring
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
Residue class ring of integers modulo 12
Residue class ring of integers modulo 12
12
12
false
false
true
true
true
true
false
true
Residue class ring of integers modulo 12
Residue class ring of integers modulo 13
RngIntResElt
RngIntResElt
9
12
0
0
true
false
false
true
false
true
false
false
false
false
true
true
12
1
>> Maximum(S) ;
^
Runtime error in 'Maximum': Bad argument types
Argument types given: RngIntRes
9
>> Minimum(R) ;
^
Runtime error in 'Minimum': Bad argument types
Argument types given: RngIntRes 本帖最后由 lilianjie1 于 2012-1-11 15:56 编辑
Z:=IntegerRing() ;Z;
I12:=ideal< Z | 12 >;
I12;
ZZ:=IntegerRing(15) ;ZZ;
IZZ15:=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，
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，
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;
I12 meet IZZ15;
I12 * IZZZ5;
I12 + IZZZ5;
I12 meet IZZZ5;
I12 / IZZZ5;
IZZZ5/ I12 ;
Z * IZZZ5;
I12 + IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;Z meet IZZZ5;
I12 meet IZZZ5;
IZZ15 meet IZZZ5;
IZZ15 / IZZZ5;
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;
Integer Ring
Ideal of Integer Ring generated by 12
Residue class ring of integers modulo 15
Ideal of Integer Ring generated by 15
false
Residue 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
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
>> I12 / IZZZ5;
^
Runtime error in '/': Argument 2 must divide argument 1.
>> IZZZ5/ I12 ;
^
Runtime error in '/': Argument 2 must divide argument 1.
Ideal of Integer Ring generated by 5
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
Ideal of Integer Ring generated by 5
Ideal of Integer Ring generated by 60
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
false
true
true
false Z:=IntegerRing() ;Z;
I12:=ideal< Z | 13 >;
I12;
ZZ:=IntegerRing(60) ;ZZ;
IZZ15:=ideal< ZZ | 31 >;
IZZ15;
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
ext< Z | > ;超越扩张到一元多项式
ext< ZZ | > ;
ext< Z, 2 | > ;超越扩张到多元多项式
ext< Z, 3 | > Completion(Z, I12) ;
comp<Z |I12 >;
素理想零理想完备化，和P进环联系起来
Completion(Z, 0) ;
comp<Z |0 >;
Integer Ring
Ideal of Integer Ring generated by 13
Residue class ring of integers modulo 60
Residue class ring of integers modulo 60
Finite field of size 13
Mapping from: RngInt: Z to GF(13)
modulo 13
>> 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
Valuation 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
Mapping from: RngInt: Z to Valuation ring of Rational Field with generator 131
Univariate Polynomial Ring over Integer Ring
Univariate Polynomial Ring over IntegerRing(60)
>> ext< Z, 2 | > ;
^
Runtime error: This constructer is no longer supported
>> ext< Z, 3 | >
^
Runtime 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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