Number Fields: Galois Groups and Class Groups
=============================================

Galois Groups
-------------

We can compute the Galois group of a number field using the ``galois_group``
function, which by default calls PARI (https://pari.math.u-bordeaux.fr/). You
do not have to worry about installing PARI, since it is part of Sage. In fact,
despite appearances much of the difficult algebraic number theory in Sage is
actually done by the PARI C library (be sure to also cite PARI in papers that
use Sage).

::

    sage: K.<alpha> = NumberField(x^6 + 40*x^3 + 1372)
    sage: G = K.galois_group()
    sage: G
    Galois group 6T2 ([3]2) with order 6 of x^6 + 40*x^3 + 1372

Internally G is represented as a group of permutations, but we can also apply
any element of G to any element of the field:

.. link

::

    sage: G.order()
    6
    sage: G.gens()
    [(1,2)(3,4)(5,6), (1,4,6)(2,5,3)]
    sage: f = G.1; f(alpha)
    1/36*alpha^4 + 1/18*alpha

Some more advanced number-theoretical tools are available via G:

.. link

::

    sage: P = K.primes_above(2)[0]
    sage: G.inertia_group(P)
    Subgroup generated by [(1,4,6)(2,5,3)] of (Galois group 6T2 ([3]2) with order 6 of x^6 + 40*x^3 + 1372)
    sage: sorted([G.artin_symbol(Q) for Q in K.primes_above(5)])  # random order, see Issue #18308
    [(1,3)(2,6)(4,5), (1,2)(3,4)(5,6), (1,5)(2,4)(3,6)]

If the number field is not Galois over `\QQ`, then the ``galois_group``
command will construct its Galois closure and return the Galois group of that:

::

    sage: K.<a> = NumberField(x^3 - 2)
    sage: G = K.galois_group(names='b'); G
    Galois group 3T2 (S3) with order 6 of x^3 - 2
    sage: G.order()
    6


Some more Galois groups
-----------------------

We compute two more Galois groups of degree :math:`5` extensions, and see that
one has Galois group :math:`S_5`, so is not solvable by radicals. For these
purposes we only need to know the structure of the Galois group as an abstract
group, rather than as an explicit group of automorphisms of the splitting
field::

    sage: NumberField(x^5 - 2, 'a').galois_group()
    Galois group 5T3 (5:4) with order 20 of x^5 - 2
    sage: NumberField(x^5 - x + 2, 'a').galois_group()
    Galois group 5T5 (S5) with order 120 of x^5 - x + 2


Magma's Galois group command
----------------------------

Recent versions of Magma have an algorithm for computing Galois groups that in
theory applies when the input polynomial has any degree. There are no open
source implementation of this algorithm (as far as I know). If you have Magma,
you can use this algorithm from Sage by calling the ``galois_group`` function
and giving the ``algorithm='magma'`` option. The return value is one of the
groups in the GAP transitive groups database.

::

    sage: K.<a> = NumberField(x^3 - 2)
    sage: K.galois_group(type="gap", algorithm='magma')  # optional - magma
    Galois group Transitive group number 2 of degree 3 of
    the Number Field in a with defining polynomial x^3 - 2

We emphasize that the above example should not work if you don't
have Magma.

Computing complex embeddings
----------------------------

You can also enumerate all complex embeddings of a number field:

.. link

::

    sage: K.complex_embeddings()
    [Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Complex Field with 53 bits of precision
       Defn: a |--> -0.629960524947437 - 1.09112363597172*I,
     Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Complex Field with 53 bits of precision
       Defn: a |--> -0.629960524947437 + 1.09112363597172*I,
     Ring morphism:
       From: Number Field in a with defining polynomial x^3 - 2
       To:   Complex Field with 53 bits of precision
       Defn: a |--> 1.25992104989487]


Class Numbers and Class Groups
------------------------------

The class group :math:`C_K` of a number field :math:`K` is the group
of fractional ideals of the maximal order :math:`R` of :math:`K`
modulo the subgroup of principal fractional ideals. One of the main
theorems of algebraic number theory asserts that :math:`C_K` is a
finite group. For example, the quadratic number field
:math:`\QQ(\sqrt{-23})` has class number :math:`3`, as we see
using the Sage ``class number`` command.

::

    sage: L.<a> = NumberField(x^2 + 23)
    sage: L.class_number()
    3


Quadratic imaginary fields with class number 1
----------------------------------------------

There are only 9 quadratic imaginary field
:math:`\QQ(\sqrt{D})` that have class number :math:`1`:

.. math::

   D = -3, -4, -7, -8, -11, -19, -43, -67, -163

To find this list using Sage, we first experiment with making lists
in Sage. For example, typing ``[1..10]`` makes the
list of integers between :math:`1` and :math:`10`.

::

    sage: [1..10]
    [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

We can also make the list of odd integers between :math:`1` and
:math:`11`, by typing ``[1,3,..,11]``, i.e., by giving the second term
in the arithmetic progression.

::

    sage: [1,3,..,11]
    [1, 3, 5, 7, 9, 11]

Applying this idea, we make the list of negative numbers from
:math:`-1` down to :math:`-10`.

::

    sage: [-1,-2,..,-10]
    [-1, -2, -3, -4, -5, -6, -7, -8, -9, -10]

Enumerating quadratic imaginary fields with class number 1
----------------------------------------------------------

The first two lines below makes a list :math:`v` of every :math:`D`
from :math:`-1` down to :math:`-200` such that :math:`D` is a
fundamental discriminant (the discriminant of a quadratic imaginary
field).

.. note::

   Note that you will not see the ... in the output below;
   this ... notation just means that part of the output is omitted
   below.

::

    sage: w = [-1,-2,..,-200]
    sage: v = [D for D in w if D.is_fundamental_discriminant()]
    sage: v
    [-3, -4, -7, -8, -11, -15, -19, -20, ..., -195, -199]

Finally, we make the list of :math:`D` in our list :math:`v` such that
the quadratic number field :math:`\QQ(\sqrt{D})` has class
number :math:`1`. Notice that ``QuadraticField(D)`` is a shorthand for
``NumberField(x^2 - D)``.

.. link

::

    sage: [D for D in v if QuadraticField(D,'a').class_number()==1]
    [-3, -4, -7, -8, -11, -19, -43, -67, -163]

Of course, we have *not* proved that this is the list of all
negative :math:`D` so that :math:`\QQ(\sqrt{D})` has
class number :math:`1`.


Class number 1 fields
---------------------

A frustrating open problem is to prove that there are infinitely many
number fields with class number :math:`1`. It is quite easy to be
convinced that this is probably true by computing a bunch of class
numbers of real quadratic fields. For example, over 58 percent of the
real quadratic number fields with discriminant :math:`D<1000` have
class number :math:`1`!

::

    sage: w = [1..1000]
    sage: v = [D for D in w if D.is_fundamental_discriminant()]
    sage: len(v)
    302
    sage: len([D for D in v if QuadraticField(D,'a').class_number() == 1])
    176
    sage: 176.0/302
    0.582781456953642

For more intuition about what is going on, read about the
Cohen-Lenstra heuristics.


Class numbers of cyclotomic fields
----------------------------------

Sage can also compute class numbers of extensions of higher degree,
within reason. Here we use the shorthand ``CyclotomicField(n)`` to
create the number field :math:`\QQ(\zeta_n)`.

::

    sage: CyclotomicField(7)
    Cyclotomic Field of order 7 and degree 6
    sage: for n in [2..15]:
    ....:     print("{} {}".format(n, CyclotomicField(n).class_number()))
    2 1
    3 1
    ...
    15 1

In the code above, the notation ``for n in [2..15]: ...`` means
"do ... for :math:`n` equal to each of the integers :math:`2,3,4,\dots,15`."

.. note::

   Exercise: Compute what is omitted (replaced by ...) in the output
   of the previous example.

Assuming conjectures to speed computations
------------------------------------------

Computations of class numbers and class groups in Sage is done by the
Pari C library, and *unlike in Pari*, by default Sage tells Pari *not
to assume* any conjectures. This can make some commands vastly slower
than they might be directly in Pari, which *does assume unproved
conjectures* by default. Fortunately, it is easy to tell Sage to be
more permissive and allow Pari to assume conjectures, either just for
this one call or henceforth for all number field functions. For
example, with ``proof=False`` it takes only a few seconds to verify,
modulo the conjectures assumed by Pari, that the class number of
:math:`\QQ(\zeta_{23})` is :math:`3`.

::

    sage: CyclotomicField(23).class_number(proof=False)
    3


.. note::

  Exercise: What is the smallest :math:`n` such that
  :math:`\QQ(\zeta_n)` has class number bigger than :math:`1`?


Class group structure
---------------------

In addition to computing class numbers, Sage can also compute the
group structure and generators for class groups. For example, the
quadratic field :math:`\QQ(\sqrt{-30})` has class group
:math:`C = (\ZZ/2\ZZ)^{\oplus 2}`, with generators the
ideal classes containing :math:`(5,\sqrt{-30})` and
:math:`(3,\sqrt{-30})`.

::

    sage: K.<a> = QuadraticField(-30)
    sage: C = K.class_group()
    sage: C
    Class group of order 4 with structure C2 x C2 of Number Field in a with defining polynomial x^2 + 30 with a = 5.477225575051661?*I
    sage: category(C)
    Category of finite enumerated commutative groups
    sage: C.gens()
    (Fractional ideal class (5, a), Fractional ideal class (3, a))


Arithmetic in the class group
-----------------------------

In Sage, the notation ``C.i`` means "the :math:`i^{th}` generator of the
object :math:`C`," where the generators are indexed by numbers
:math:`0, 1, 2, \dots`. Below, when we write ``C.0 \* C.1``, this
means "the product of the 0th and 1st generators of the class group
:math:`C`."

::

    sage: K.<a> = QuadraticField(-30)
    sage: C = K.class_group()
    sage: C.0
    Fractional ideal class (5, a)
    sage: C.0.ideal()
    Fractional ideal (5, a)
    sage: I = C.0 * C.1
    sage: I
    Fractional ideal class (2, a)


Next we find that the class of the fractional ideal
:math:`(2,\sqrt{-30}+4/3)` is equal to the ideal class
:math:`C.0*C.1`.

.. link

::

    sage: A = K.ideal([2, a+4/3])
    sage: J = C(A)
    sage: J
    Fractional ideal class (2/3, 1/3*a)
    sage: J == C.0*C.1
    True


Unfortunately, there is currently no Sage function that writes a
fractional ideal class in terms of the generators for the class
group.
