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 (http://pari.math.u-bordeaux.fr/). You do not have to worry about installing Pari, since Pari 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 of Number Field in alpha with defining polynomial 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:

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:

sage: P = K.primes_above(2)[0]
sage: G.inertia_group(P)
Subgroup [(), (1,4,6)(2,5,3), (1,6,4)(2,3,5)] of Galois group of Number Field in alpha with defining polynomial x^6 + 40*x^3 + 1372
sage: sorted([G.artin_symbol(Q) for Q in K.primes_above(5)])
[(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; you need to give it a variable name for the generator of the Galois closure:

sage: K.<a> = NumberField(x^3 - 2)
sage: G = K.galois_group(names='b'); G
Galois group of Galois closure in b of Number Field in a with defining polynomial x^3 - 2
sage: G.order()
6

Some more Galois groups

We compute two more Galois groups of degree \(5\) extensions, and see that one has Galois group \(S_5\), so is not solvable by radicals. For these purposes we only want to know the structure of the Galois group as an abstract group, rather than as an explicit group of automorphisms of the splitting field; this is much quicker to calculate. PARI has a type for representing “abstract Galois groups”, and Sage can use this.:

sage: NumberField(x^5 - 2, 'a').galois_group(type="pari")
Galois group PARI group [20, -1, 3, "F(5) = 5:4"] of
degree 5 of the Number Field in a with defining
polynomial x^5 - 2
sage: NumberField(x^5 - x + 2, 'a').galois_group(type="pari")
Galois group PARI group [120, -1, 5, "S5"] of degree 5 of
the Number Field in a with defining polynomial 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
verbose...
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:

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 \(C_K\) of a number field \(K\) is the group of fractional ideals of the maximal order \(R\) of \(K\) modulo the subgroup of principal fractional ideals. One of the main theorems of algebraic number theory asserts that \(C_K\) is a finite group. For example, the quadratic number field \(\QQ(\sqrt{-23})\) has class number \(3\), 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 \(\QQ(\sqrt{D})\) that have class number \(1\):

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

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 \(1\) and \(10\).

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

We can also make the list of odd integers between \(1\) and \(11\), 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 \(-1\) down to \(-10\).

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 \(v\) of every \(D\) from \(-1\) down to \(-200\) such that \(D\) 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 is_fundamental_discriminant(D)]
sage: v
[-3, -4, -7, -8, -11, -15, -19, -20, ..., -195, -199]

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

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 \(D\) so that \(\QQ(\sqrt{D})\) has class number \(1\).

Class number 1 fields

A frustrating open problem is to prove that there are infinitely many number fields with class number \(1\). 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 \(D<1000\) have class number \(1\)!

sage: w = [1..1000]
sage: v = [D for D in w if is_fundamental_discriminant(D)]
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 \(\QQ(\zeta_n)\).

sage: CyclotomicField(7)
Cyclotomic Field of order 7 and degree 6
sage: for n in [2..15]: print 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 \(n\) equal to each of the integers \(2,3,4,\dots,15\).”

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 \(\QQ(\zeta_{23})\) is \(3\).

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

Note

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

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 \(\QQ(\sqrt{-30})\) has class group \(C = (\ZZ/2\ZZ)^{\oplus 2}\), with generators the ideal classes containing \((5,\sqrt{-30})\) and \((3,\sqrt{-30})\).

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
sage: category(C)
Category of finite commutative groups
sage: C.gens()
(Fractional ideal class (2, a), Fractional ideal class (3, a))

Arithmetic in the class group

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

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

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

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
True

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