# Constructing Artin Representations ## `ArtinRepresentations(K): FldNum -> SeqEnum` ```magma 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 $K$. If $K$ is given as a relative extension $E/F$ (that is, `BaseField(K)` is not ${\mathbb{Q}}$), the representations returned are those of ${\operatorname{Gal}}(\bar F/F)$ that factor through the normal closure of $E$ over $F$, and `LSeries` returns the corresponding Artin $L$-series over $F$. To obtain the representations of ${\operatorname{Gal}}(\bar{\mathbb{Q}}/{\mathbb{Q}})$ instead, apply `ArtinRepresentations` to `AbsoluteField(K)`. The Galois group $G={\operatorname{Gal}}(F/K)$ 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 ${\mathbb{Q}}$. (It is possible to specify any monic integral polynomial whose splitting field is $F$, even a reducible one, but `PermutationCharacter(K)` and the Dedekind $\zeta$-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:=` `` 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 β€œ(?)” 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, AlgChtrElt -> ArtRep` ## `K !! ch: FldNum, SeqEnum -> ArtRep` Writing $F$ for the normal closure of $K/{\mathbb{Q}}$, this function converts an abstract group character of ${\operatorname{Gal}}(F/{\mathbb{Q}})$ or the sequence of its trace values into an Artin representation. ## `PermutationCharacter(K): FldNum -> ArtRep` Construct the permutation representation $A$ of the absolute Galois group of ${\mathbb{Q}}$ on the embeddings of $K$ into ${\mathbb{C}}$. This is an Artin representation of ${\operatorname{Gal}}(F/{\mathbb{Q}})$ of dimension $[K:{\mathbb{Q}}]$, where $F$ is the normal closure of $K$, and it is the same as the permutation representation of ${\operatorname{Gal}}(F/{\mathbb{Q}})$ on the cosets of ${\operatorname{Gal}}(F/K)$. ## `Determinant(A): ArtRep -> ArtRep` Construct the determinant of a given Artin representation. The result is given as a 1-dimensional Artin representation attached to the same field. ## `ChangeField(A, K): ArtRep, FldNum -> ArtRep, BoolElt` ## `K !! A: FldNum, ArtRep -> ArtRep, BoolElt` ```magma 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 β€œthe 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. ## `Example: Artin Const (ex-81e6bd)` A quadratic field $K$ has two irreducible Artin representations the factor through ${\operatorname{Gal}}(K/{\mathbb{Q}})$, the trivial one and the quadratic character of $K$: ```magma > K := 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: ```magma > triv,sign:Magma; QuadraticField(-1) !! [1,1] QuadraticField(-1) !! [1,-1] ``` The regular representation of ${\operatorname{Gal}}(K/{\mathbb{Q}})$ is their sum: ```magma > PermutationCharacter(K); Artin representation C2: (2,0) of Q(sqrt(-1)) > $1 eq triv+sign; true ``` Next, let $L=K(\sqrt{-2-i})$, presented as a relative extension of $K$. As $L$ is defined over $K$, `ArtinRepresentations` returns the representations of ${\operatorname{Gal}}(\bar K/K)$, here the trivial character and the quadratic character of ${\operatorname{Gal}}(L/K)$: ```magma > P := PolynomialRing(K); > L := ext; > reps := ArtinRepresentations(L); > [Dimension(A): A in reps]; [ 1, 1 ] > BaseField(reps[2]) eq K; true ``` The normal closure $F$ of $L$ over ${\mathbb{Q}}$ has ${\operatorname{Gal}}(F/{\mathbb{Q}})=D_4$, the dihedral group of order 8. Passing to the absolute field instead gives the Artin representations of ${\operatorname{Gal}}(\bar{\mathbb{Q}}/{\mathbb{Q}})$: ```magma > Labs := AbsoluteField(L); > GroupName(GaloisGroup(Labs)); D4 > [Dimension(A): A in ArtinRepresentations(Labs)]; [ 1, 1, 1, 1, 2 ] ``` We use `ChangeField` to lift Artin representations from ${\operatorname{Gal}}(K/{\mathbb{Q}})$ to ${\operatorname{Gal}}(F/{\mathbb{Q}})$, and check that it is still the same as an Artin representation. ```magma > A := ChangeField(sign,L); > A; Artin representation D4: (1,1,-1,1,-1) of ext > A eq sign; true ```