abstract-algebra.rst

.. -*- coding: utf-8 -*-

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.. _prep-quickstart-abstract-algebra:

Sage Quickstart for Abstract Algebra
====================================

This `Sage <http://www.sagemath.org>`_ quickstart tutorial was developed
for the MAA PREP Workshop "Sage: Using Open\-Source Mathematics Software
with Undergraduates" (funding provided by NSF DUE 0817071).  It is
licensed under the Creative Commons Attribution\-ShareAlike 3.0 license
(`CC BY\-SA <http://creativecommons.org/licenses/by-sa/3.0/>`_).

As computers are discrete and finite, anything with a discrete, finite
set of generators is natural to implement and explore.

Group Theory
-------------

Many common groups are pre-defined, usually as permutation groups:
that is, explicitly described as subgroups of symmetric groups.

- Every group of order 15 or less is available as a permutation group.

- Sometimes they are available under special names, though.

::

    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

We can access a lot of information about groups, such as:

- A list of subgroups up to conjugacy,

- or a stabilizer,

- or other things demonstrated below.

::

    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 [(1,5)(2,6)(3,7)(4,8), (1,3,5,7)(2,4,6,8)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (2,8)(3,7)(4,6)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (1,2)(3,8)(4,7)(5,6)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (1,3,5,7)(2,4,6,8), (2,8)(3,7)(4,6)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (1,3,5,7)(2,4,6,8), (1,2,3,4,5,6,7,8)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (1,3,5,7)(2,4,6,8), (1,2)(3,8)(4,7)(5,6)] of (Dihedral group of order 16 as a permutation group)
    Subgroup generated by [(1,5)(2,6)(3,7)(4,8), (1,3,5,7)(2,4,6,8), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8)] of (Dihedral group of order 16 as a permutation group)

In the previous cell we once again did a for loop over a set of objects
rather than just a list of numbers.  This can be very powerful.

::

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

::

    sage: len(D.normal_subgroups())
    7
    sage: for K in sorted(D.normal_subgroups()):
    ....:     print(K)
    Subgroup generated by [()] of (Dihedral group of order 16 as a permutation group)
    ...
    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)

We can access specific subgroups if we know the generators 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

There are some matrix groups as well, both finite and infinite.

::

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

We can print out *all* of the elements of this group.

::

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

::

    sage: SS = SL(2, ZZ)

Of course, you have to be careful what you try to do!

::

    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]
    ...

Rings
------

Sage has many pre\-defined rings to experiment with.  Here is how one
would access :math:`\ZZ/12\ZZ`, for instance.

::

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

::

    sage: twelve.is_field()
    False

::

    sage: twelve.is_integral_domain()
    False

Quaternions, and generalizations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We can define generalized quaternion algebras, where :math:`i^2=a`,
:math:`j^2=b`, and :math:`k=i\cdot j`, all over :math:`\QQ`::

    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

Polynomial Rings
~~~~~~~~~~~~~~~~

Polynomial arithmetic in Sage is a very important tool.

The first cell brings us back to the symbolic world.  This is **not the
same thing** as polynomials!

::

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

Now we will turn :math:`x` into the generator of a polynomial ring.  The
syntax is a little unusual, but you will see it often.

::

    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

Fields
------

Sage has superb support for finite fields and extensions of the rationals.

Finite Fields
~~~~~~~~~~~~~

::

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

The generator satisfies a Conway polynomial, by default, or the
polynomial can be specified.

::

    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

:math:`F` should be the splitting field of the polynomial
:math:`x^{81}-x`, so it is very good that we get no output from the
following cell, which combines a loop and a conditional statement.

::

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

Field Extensions, Number Fields
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Most things you will need in an undergraduate algebra classroom are
already in Sage.

::

    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