f: RngUPolElt Default:
Ramification: BoolElt Default: false
FactorDiscriminant: BoolElt Default: false
p0: RngIntElt Default:
Compute all irreducible Artin representations that factor through the normal
closure F of the number field K.
The Galois group G=Gal(F/K) whose representations are constructed is
represented as a permutation group on the roots of f, which must be
a monic irreducible polynomial with integer coefficients that defines K.
By default this is the defining polynomial of K
represented as an extension of Q.
(It is possible to specify any monic integral polynomial
whose splitting field is F, even a reducible one,
but PermutationCharacter(K) and the Dedekind ζ-function of K
will not work correctly.)
The Ramification parameter specifies whether to pre-compute the inertia
groups at all ramified primes and the conductors of all representations.
The parameter FactorDiscriminant determines whether to factorize the discriminant
of f completely, even if it appears to contain large prime factors.
The factorization is used to determine which primes ramify in F/K,
which is necessary to compute the conductors. If the factorization is
incomplete, Magma assumes that the primes
in the unfactored part of the discriminant are unramified. One may specify
FactorDiscriminant:= <TrialLimit,PollardRhoLimit,ECMLimit,MPQSLimit,Proof> and these 5 parameters are passed
to the Factorization function; the default behaviour (false)
is the same as <10000,65535,10,0,false>.
When the factorization is incomplete, Magma will print "(?)"
following the conductor values, when asked to print an Artin representation.
Finally, p0 specifies which p-adic field to use for the roots of
f, in particular in Galois group computations.
It must be chosen so that GaloisGroup(f:Prime:=p0) is successful.
By default it is chosen by the Galois group computation.
K !! ch : FldNum, SeqEnum -> ArtRep
Writing F for the normal closure of K/Q, this function converts
an abstract group character of Gal(F/Q) or the sequence of its trace
values into an Artin representation.
Construct the permutation representation A of the absolute Galois group of
Q on the embeddings of K into C. This is an Artin representation
of Gal(F/Q) of dimension [K:Q], where
F is the normal closure of K, and it is the same as the permutation
representation of Gal(F/Q) on the cosets of Gal(F/K).
Construct the determinant of a given Artin representation. The result
is given as a 1-dimensional Artin representation attached to the same field.
K !! A : FldNum, ArtRep -> ArtRep, BoolElt
MinPrimes: RngIntElt Default: 20
Given an Artin representation (attached to some number field) that is known
to factor through
the Galois closure of K, attempts to recognize it as such.
Returns "the resulting Artin representation attached to K", true
if successful, and 0, false
if it proves that there is no such representation.
The parameter MinPrimes specifies the number of additional primes
for which to compare traces of Frobenius elements.
A quadratic field K has two irreducible Artin representations the factor
through Gal(K/Q), the trivial one and the quadratic character of K:
> K<i> := QuadraticField(-1);
> triv, sign := Explode(ArtinRepresentations(K));
> sign;
Artin representation C2: (1,-1) of Q(sqrt(-1))
An alternative way to define them is directly by their character:
> triv,sign:Magma;
QuadraticField(-1) !! [1,1]
QuadraticField(-1) !! [1,-1]
The regular representation of Gal(K/Q) is their sum:
> PermutationCharacter(K);
Artin representation C2: (2,0) of Q(sqrt(-1))
> $1 eq triv+sign;
true
Next, let L=K(Sqrt( - 2 - i)). Then L has normal
closure F with Gal(F/Q)=D 4, the dihedral group of order 8:
> L := ext<K|Polynomial([2+i,0,1])>;
> G := GaloisGroup(AbsoluteField(L));
> GroupName(G);
D4
> [Dimension(A): A in ArtinRepresentations(L)];
[ 1, 1, 1, 1, 2 ]
We use ChangeField to lift Artin representations from Gal(K/Q) to
Gal(F/Q), and check that it is still the same as an Artin representation.
> A := ChangeField(sign,L);
> A;
Artin representation D4: (1,1,-1,1,-1) of ext<Q(sqrt(-1))|x^2+i+2>
> A eq sign;
true
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