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The group of relative automorphisms of the abelian extension is isomorphic
via the Artin map to the ideal group used to define the field.
After defining equations are computed, the user can explicitly map ideals
that are coprime to the defining modulus to automorphisms of the field.
Returns a map from the defining group (considered as a "subgroup" of the
ideals of the base ring) into the automorphisms of the abelian extension A
over the base field.
By the defining property of class fields, this map induces an isomorphism
on the defining group (as an abelian group) onto the relative
automorphisms of A.
Since this function constructs the number field defined by A, this may
involve a lengthy calculation.
Computes the relative automorphism of the abelian extension A that is the
Frobenius automorphism of p.
Since this function constructs the number field defined by A, this may
involve a lengthy calculation.
All: BoolElt Default: false
Over: [Map] Default: [ ]
If IsNormal is true for the abelian extension A with the given
parameters, then the
automorphism group of A over k0 is computed.
Since this function constructs the number field defined by A, this may
involve a lengthy calculation.
Factor: RingIntElt Default: 1
In case of A and it's base field k both begin normal over Q,
the automorphism group G of A/Q is a group extension of the
abelian group coming from the definition of A and the automorphism
group of k/Q. This functions sets up the corresponding group
extension problem and uses DistinctExtensions to compute
all group theoretical possibilities for G. In case of several
possible groups, a further selection based on cycle types and their
frequencies is attempted. The optional parameter Factor is
passed on to ImproveAutomorphismGroup to control
the amount of time spent on improving the guess.
While this function can be much faster than the direct computation of
the automorphism group, the result of this computation is in general
not guaranteed. Furthermore, as there are groups that cannot be
distinguished by cycle types and their frequencies alone, correctness
cannot be achieved by increasing the value of Factor.
The intended use of this function is to have a (reasonable fast)
method of checking is the field under consideration has an interesting
group before an unnecessary long call to AutomorphismGroup is
attempted.
Factor: RngIntElt Default: 1
Given the output of ProbableAutomorphismGroup or
ImproveAutomorphismGroup try to improve the quality of the guess
by splitting more primes to get more data for a cycle-type frequency
analysis.
We will demonstrate the use of ProbableAutomorphismGroup
by investigating extensions of Q(sqrt (10)):
> k := NumberField(Polynomial([-10, 0, 1]));
> R, mR := RayClassGroup(4*3*5*MaximalOrder(k));
> s := [x`subgroup : x in Subgroups(R:Quot := [2,2])];
> a := [ AbelianExtension(Inverse(mq)*mR) where
> _, mq := quo<R|x> : x in s ];
> n := [ x : x in a | IsNormal(x:All)];
> ProbableAutomorphismGroup(n[2]);
Finitely presented group on 3 generators
Relations
$.2^2 = Id($)
$.3^2 = Id($)
($.2, $.3) = Id($)
($.1, $.2^-1) = Id($)
$.1^-1 * $.3 * $.1 * $.3^-1 * $.2^-1 = Id($)
$.1^2 = Id($)
This shows that since there is only one group extension of a V4
by C2 with the action induced from the action of the Galois group
of k on R, the Automorphism group is already determined.
On the other hand, for the first subgroup there are more possibilities:
> g, c := ProbableAutomorphismGroup(n[1]);
> #c;
2
We will try to find the "correct" guess by looking at the orders
of elements which correspond to decomposition types and their frequencies:
> {* Order(x) : x in CosetImage(c[1], sub<c[1]|>) *};
{* 1, 2^^7 *}
> {* Order(x) : x in CosetImage(c[2], sub<c[2]|>) *};
{* 1, 2^^3, 4^^4 *}
So, if we find a prime of degree 4 we know it's the second group.
Looking at the frequencies, we can be pretty confident that we should
be able to find a suitable prime - if it exists. Since among the
first 100 primes there is not a single prime with a factor of degree
4 we are pretty confident that the first group is the correct one.
By setting the verbose level, we can see how the decision is made:
> SetVerbose("ClassField", 2);
> _ := ImproveAutomorphismGroup(n[1], c:Factor := 2);
Orders and multiplicities are [
{* 1, 2^^7 *},
{* 1, 2^^3, 4^^4 *}
]
Probable orders and multiplicities are {* 1^^3, 2^^34 *}
Error terms are [ 0.00699329636679632541587354133907,
0.993006703633203674584126458661 ]
This indicates that out of 37 primes considered, non had a degree 4
factor, thus we are confident that the first group is the correct one.
Given an abelian extension, compute it's Galois group over Q,
ie. the abstract automorphism group of a Q-normal closure of A.
This function requires the defining equations for A as a number field
to be known, but is considerably faster than calling GaloisGroup
for the number field directly.
The group is returned as an permutation group. All roots of the
defining polynomial of the coefficient field of A as well as of the
defining polynomials of A itself are returned in some local field.
The zeros as well as data required for further computations
are contained in the 3rd return value.
For an abelian extension that is normal over Q and defined over a
normal base field k/Q, the automorphism group of A/Q is a
group extension of the Galois group of k by A. As a group extension
it corresponds to an element in the second cohomology group and can be
represented by an explicit 2-cocyle with values in the norm group.
This function computes such a cocycle. It can be used as an element
of the second cohomology group of the cohomology module of A, see
CohomologyModule.
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