# 抽象代数的Sage快速入门¶

## 群论¶

• 每一个15阶或更低阶的组都可以作为置换组使用。

• 不过，有时它们也有特殊的名称。

sage: G = QuaternionGroup()
sage: G
Quaternion group of order 8 as a permutation group

sage: H = AlternatingGroup(5)
sage: H
Alternating group of order 5!/2 as a permutation group

sage: H.is_simple()
True

sage: D = DihedralGroup(8)
sage: D
Dihedral group of order 16 as a permutation group


• 直到共轭的子群的列表，

• 或者稳定剂，

• 或者下面展示的其他东西。

sage: for K in D.conjugacy_classes_subgroups():
....:     print(K)
Subgroup generated by [()] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(2,8)(3,7)(4,6)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,2)(3,8)(4,7)(5,6)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(2,8)(3,7)(4,6), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,2)(3,8)(4,7)(5,6), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(2,8)(3,7)(4,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,2)(3,8)(4,7)(5,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)


sage: D.stabilizer(3)
Subgroup generated by [(1,5)(2,4)(6,8)] of (Dihedral group of order 16 as a permutation group)

sage: for K in D.normal_subgroups():
....:     print(K)
Subgroup generated by [(1,2,3,4,5,6,7,8), (1,8)(2,7)(3,6)(4,5)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,2,3,4,5,6,7,8), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,8)(2,7)(3,6)(4,5)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(2,8)(3,7)(4,6), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [(1,5)(2,6)(3,7)(4,8)] of (Dihedral group of order 16 as a permutation group)
Subgroup generated by [()] of (Dihedral group of order 16 as a permutation group)


sage: L = D.subgroup(["(1,3,5,7)(2,4,6,8)"])

sage: L.is_normal(D)
True

sage: Q=D.quotient(L)
sage: Q
Permutation Group with generators [(1,2)(3,4), (1,3)(2,4)]

sage: Q.is_isomorphic(KleinFourGroup())
True


sage: S = SL(2, GF(3))
sage: S
Special Linear Group of degree 2 over Finite Field of size 3


sage: for a in S:
....:     print(a)
[1 0]
[0 1]
...
[2 2]
[2 1]

sage: SS = SL(2, ZZ)


sage: SS.list()
Traceback (most recent call last):
...
NotImplementedError: group must be finite

sage: for a in SS.gens():
....:     print(a)
[ 0  1]
[-1  0]
...


## 环¶

Sage有许多预定义的环可供实验。下面是一个如何访问 $$\ZZ/12\ZZ$$ 例如。

sage: twelve = Integers(12)
sage: twelve
Ring of integers modulo 12

sage: twelve.is_field()
False

sage: twelve.is_integral_domain()
False


### 四元数和推广¶

sage: quat = QuaternionAlgebra(-1, -1)
sage: quat
Quaternion Algebra (-1, -1) with base ring Rational Field

sage: quat.is_field()
False

sage: quat.is_commutative()
False

sage: quat.is_division_algebra()
True

sage: quat2 = QuaternionAlgebra(5, -7)

sage: quat2.is_division_algebra()
True

sage: quat2.is_field()
False


### 多项式环¶

sage: reset('x') # This returns x to being a variable
sage: (x^4 + 2*x).parent()
Symbolic Ring


sage: R.<x> = QQ[]
sage: R
Univariate Polynomial Ring in x over Rational Field

sage: R.random_element() # random
-5/2*x^2 - 1/4*x - 1

sage: R.is_integral_domain()
True

sage: (x^4 + 2*x).parent()
Univariate Polynomial Ring in x over Rational Field

sage: (x^2+x+1).is_irreducible()
True

sage: F = GF(5)
sage: P.<y> = F[]

sage: P.random_element() # random
2*y

sage: I = P.ideal(y^3+2*y)
sage: I
Principal ideal (y^3 + 2*y) of Univariate Polynomial Ring in y over Finite Field of size 5

sage: Q = P.quotient(I)

sage: Q
Univariate Quotient Polynomial Ring in ybar over Finite Field of size 5 with modulus y^3 + 2*y


## 领域¶

Sage对有限域和有理数的扩张有极好的支持。

### 有限域¶

sage: F.<a> = GF(3^4)
sage: F
Finite Field in a of size 3^4


sage: F.polynomial()
a^4 + 2*a^3 + 2

sage: F.list()
[0, a, a^2, a^3, a^3 + 1, a^3 + a + 1, a^3 + a^2 + a + 1, 2*a^3 + a^2 + a + 1, a^2 + a + 2, a^3 + a^2 + 2*a, 2*a^3 + 2*a^2 + 1, a^3 + a + 2, a^3 + a^2 + 2*a + 1, 2*a^3 + 2*a^2 + a + 1, a^3 + a^2 + a + 2, 2*a^3 + a^2 + 2*a + 1, 2*a^2 + a + 2, 2*a^3 + a^2 + 2*a, 2*a^2 + 2, 2*a^3 + 2*a, 2*a^3 + 2*a^2 + 2, a^3 + 2*a + 2, a^3 + 2*a^2 + 2*a + 1, 2*a^2 + a + 1, 2*a^3 + a^2 + a, a^2 + 2, a^3 + 2*a, a^3 + 2*a^2 + 1, a + 1, a^2 + a, a^3 + a^2, 2*a^3 + 1, 2*a^3 + a + 2, 2*a^3 + a^2 + 2*a + 2, 2*a^2 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a, a^3 + 2*a^2 + 2, 2*a + 1, 2*a^2 + a, 2*a^3 + a^2, 2, 2*a, 2*a^2, 2*a^3, 2*a^3 + 2, 2*a^3 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a + 2, a^3 + 2*a^2 + 2*a + 2, 2*a^2 + 2*a + 1, 2*a^3 + 2*a^2 + a, a^3 + a^2 + 2, 2*a^3 + 2*a + 1, 2*a^3 + 2*a^2 + a + 2, a^3 + a^2 + 2*a + 2, 2*a^3 + 2*a^2 + 2*a + 1, a^3 + 2*a^2 + a + 2, a^2 + 2*a + 1, a^3 + 2*a^2 + a, a^2 + 1, a^3 + a, a^3 + a^2 + 1, 2*a^3 + a + 1, 2*a^3 + a^2 + a + 2, a^2 + 2*a + 2, a^3 + 2*a^2 + 2*a, 2*a^2 + 1, 2*a^3 + a, 2*a^3 + a^2 + 2, 2*a + 2, 2*a^2 + 2*a, 2*a^3 + 2*a^2, a^3 + 2, a^3 + 2*a + 1, a^3 + 2*a^2 + a + 1, a^2 + a + 1, a^3 + a^2 + a, 2*a^3 + a^2 + 1, a + 2, a^2 + 2*a, a^3 + 2*a^2, 1]

sage: (a^3 + 2*a^2 + 2)*(2*a^3 + 2*a + 1)
2*a^3 + a^2 + a + 1


$$F$$ 应该是多项式的分裂域 $$x^{{81}}-x$$ ，所以很好的情况是我们没有从下面的单元中得到输出，这个单元组合了一个循环和一个条件语句。

sage: for a in F:
....:     if not (a^81 - a == 0):
....:         print("Oops!")


### 字段扩展名，数字字段¶

sage: N = QQ[sqrt(2)]
sage: N
Number Field in sqrt2 with defining polynomial x^2 - 2 with sqrt2 = 1.414213562373095?

sage: var('z')
z
sage: M.<a>=NumberField(z^2-2)
sage: M
Number Field in a with defining polynomial z^2 - 2

sage: M.degree()
2

sage: M.is_galois()
True

sage: M.is_isomorphic(N)
True