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.
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.
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.
> 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)=D4, 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