# Automorphism Groups Automorphisms of an algebraic field and the group they form can be calculated. Furthermore, field invariants that relate to the automorphism group can be determined. ## `Automorphisms(F): FldAlg -> [ Map ]` ```magma Abelian: BoolElt Default: false verbose: AutomorphismGroup Default: Verbose : 3 ``` Given an algebraic field $F$, return the automorphisms of $F$ as a sequence of maps. If the extension is known to be abelian, the parameter `Abelian` should be set to `true` in which case a much more efficient algorithm [[Acciaro and Klüners, 1999](../../references.md#cite-ackl1), [Klüners, 1997](../../references.md#cite-kl2)] will be employed. If $F$ is not a normal extension, the automorphisms are obtained by a variation of the polynomial factorisation algorithm. ## `AutomorphismGroup(F): FldAlg -> GrpPerm, PowMap, Map` ```magma Abelian : BoolElt Default: false UseGalois: BoolElt Default: false verbose : AutomorphismGroup Default: Verbose : 3 ``` Given an algebraic field $F$, that is either a simple normal extension of ${\mathbb{Q}}$ or simple abelian extension of ${\mathbb{Q}}$, return the automorphism group $G$ of $K$ as a permutation group of degree $n$, where $n$ is the degree of the extension. If the extension is known to be abelian, the parameter `Abelian` should be set to `true` in which case a much more efficient algorithm [[Acciaro and Klüners, 1999](../../references.md#cite-ackl1), [Klüners, 1997](../../references.md#cite-kl2)] will be employed. If `UseGalois` is set, `GaloisGroup` will be called and the result will be derived from that. If $F$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. In addition to returning $G$, the function also returns the power structure $Aut$ of all automorphisms of $F$, and the transfer map $\phi$ from $G$ into $Aut$. ## `Example: Automorphisms (ex-7b9102)` We consider the extension obtained by adjoining a root of the irreducible polynomial $x^4 - 4x^2 + 1$ to ${\mathbb{Q}}$. ```magma > Q := RationalField(); > R := PolynomialRing(Q); > K := NumberField(x^4 - 4*x^2 + 1); > A := Automorphisms(K); > A; [ Mapping from: FldNum: K to FldNum: K, Mapping from: FldNum: K to FldNum: K, Mapping from: FldNum: K to FldNum: K, Mapping from: FldNum: K to FldNum: K ] > for phi in A do phi(w); end for; w w^3 - 4*w -w^3 + 4*w -w ``` Taking the same field $K$ we use instead the function `AutomorphismGroup`: ```magma > G, Aut, tau := AutomorphismGroup(K); > for x in G do tau(x)(w); end for; w w^3 - 4*w -w^3 + 4*w -w ``` ## `AutomorphismGroup(K, F): FldAlg, FldAlg -> GrpPerm, PowMap, Map` ## `AutomorphismGroup(K, F): FldRat, FldAlg -> GrpPerm, PowMap, Map` ## `AutomorphismGroup(K, F): FldAlg, FldRat -> GrpPerm, PowMap, Map` Computes the group of $K$ automorphisms of $F$ as a permutation group together with a list of all automorphisms and a map between the permutation group and explicit automorphisms of the field. This function computes the automorphism group of $F$ over ${\mathbb{Q}}$ first. ## `DecompositionGroup(p): RngIntElt -> GrpPerm` ## `DecompositionGroup(p): RngOrdIdl -> GrpPerm` For an ideal $p$ of the maximal order of some absolute normal field $F$ with group of automorphisms $G$, compute the decomposition group, i.e. the subgroup $U$ of the automorphism group such that: $$ U := \{ s \in G | s(p) = p \} $$ If $F$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `RamificationGroup(p, i): RngOrdIdl, RngIntElt -> GrpPerm` For an ideal $p$ of the maximal order $M$ of some absolute normal field $F$ with group of automorphisms $G$, compute the $i$-th ramification group, i.e. the subgroup $U$ of the automorphism group such that: $$ U := \{ s \in G | s(x) - x \in p^{i+1} \text{for all }x\text{ in }M \} $$ If $F$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `RamificationGroup(p): RngOrdIdl -> GrpPerm` This is just an abbreviation for `RamificationGroup(p, 1)`. ## `InertiaGroup(p): RngOrdIdl -> GrpPerm` This is just an abbreviation for `RamificationGroup(p, 0)`. ## `FixedField(K, U): FldAlg, GrpPerm -> FldNum, Map` Given a normal field $K$ over ${\mathbb{Q}}$ and a subgroup $U$ of `AutomorphismGroup(K)`, this returns the largest subfield of $K$ that is fixed by $U$. This function is inverse to [`FixedGroup`](#function-rngord-fixedgroup). If $K$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `FixedField(K, S): FldAlg, [Map] -> FldAlg, Map` Given a field $K$ and a sequence of automorphisms of $K$, this returns the largest subfield of $K$ that is fixed by the given automorphisms. ## `FixedGroup(K, L): FldAlg, FldAlg -> GrpPerm` Given a normal field $K$ over ${\mathbb{Q}}$ and a subfield $L$, compute the subgroup $U$ of the `AutomorphismGroup(K)` that fixes $L$. This function is inverse to [`FixedField`](#function-rngord-fixedfield). If $K$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `FixedGroup(K, L): FldAlg, [FldAlgElt] -> GrpPerm` Given a normal field $K$ over ${\mathbb{Q}}$ and a sequence of number field elements $L$, compute the subgroup $U$ of the `AutomorphismGroup(K)` that fixes $L$. If $K$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `FixedGroup(K, a): FldAlg, FldAlgElt -> GrpPerm` Given a normal field $K$ over ${\mathbb{Q}}$ and a number field element $a$, compute the subgroup $U$ of the `AutomorphismGroup(K)` that fixes $a$. This function is inverse to [`FixedField`](#function-rngord-fixedfield). If $K$ is not a normal extension of ${\mathbb{Q}}$ an error will occur. ## `DecompositionField(p): RngOrdIdl -> FldNum, Map` This is an abbreviation for `FixedField(K, DecompositionGroup(p))` where $K$ is the number field of the order of $p$. ## `RamificationField(p, i): RngOrdIdl, RngIntElt -> FldNum, Map` This is an abbreviation for `FixedField(K, RamificationGroup(p, i))` where $K$ is the number field of the order of $p$. ## `RamificationField(p): RngOrdIdl -> FldNum, Map` This is an abbreviation for `FixedField(K, RamificationGroup(p))` where $K$ is the number field of the order of $p$. ## `InertiaField(p): RngOrdIdl -> FldNum, Map` This is an abbreviation for `FixedField(K, InertiaField(p))` where $K$ is the number field of the order of $p$. ## `Example: Ramification (ex-3b0e83)` We will demonstrate the various groups and fields. In order to do so, we first construct a non-trivial normal field. ```magma > o := MaximalOrder(ext.1^4-3); > os := MaximalOrder(SplittingField(NumberField(o))); > P := Decomposition(os, 2)[1][1]; > G, M := RayClassGroup(P^3); > G; Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*G.1 = 0 ``` Since $G$ is cyclic and the module $P$ invariant under the automorphisms of $os$, the class field corresponding to $G$ will be normal over $Q$. It Galois group over $Q$ will be an extension of $D_4$ by $C_2$. ```magma > A := AbelianExtension(M); > O := MaximalOrder(EquationOrder(A)); > Oa := AbsoluteOrder(O); > Ka := NumberField(Oa); > Gal, _, Map := AutomorphismGroup(Ka); > Gal; Permutation group Gal acting on a set of cardinality 16 Order = 16 = 2^4 (1, 2, 7, 5)(3, 8, 6, 10)(4, 12, 14, 9)(11, 16, 13, 15) (1, 3, 7, 6)(2, 8, 5, 10)(4, 13, 14, 11)(9, 16, 12, 15) (1, 4)(2, 9)(3, 11)(5, 12)(6, 13)(7, 14)(8, 15)(10, 16) ``` Now, let us pick some ideals. The only interesting primes are the primes dividing the discriminant, which in this case will be the primes over $2$ and $3$. ```magma > P2 := Decomposition(Oa, 2)[1][1]; > P3 := Decomposition(Oa, 3)[1][1]; ``` First, the valuation of the different of $Oa$ at $P2$ should be $\sum_{i=0}^\infty (\#G(P2, i)-1)$ where $G(P2, i)$ is the $i$-th ramification group. ```magma > s := 0; i := 0; > repeat > G := RamificationGroup(P2, i); > s +:= #G-1; > print i, "-th ramification group is of order ", #G; > i +:= 1; > until #G eq 1; 0 -th ramification group is of order 8 1 -th ramification group is of order 8 2 -th ramification group is of order 2 3 -th ramification group is of order 2 4 -th ramification group is of order 2 5 -th ramification group is of order 2 6 -th ramification group is of order 1 > s; 18 > Valuation(Different(Oa), P2); 18 ``` According to the theory, $P2$ should be totally ramified over the inertia field and unramified over $Q$: ```magma > K2 := InertiaField(P2); > M2 := MaximalOrder(K2); > K2r := RelativeField(K2, Ka); > M2r := MaximalOrder(K2r); > p2 := M2 meet (MaximalOrder(K2r)!!P2); > IsInert(p2); true > IsTotallyRamified(M2r!!P2); true ``` Now we try the same for $P3$. Since $3$ is split in $Ka$, we may consider an additional field: the decomposition field. It should be the maximal subfield if $K$ such that $3$ is neither inert ($f=1$) nor ramified ($e=1$), therefore $3$ has to split totally. ```magma > D3 := DecompositionField(P3); > D3M := MaximalOrder(D3); > IsTotallySplit(3, D3M); true ``` The inertia field is the maximal subfield such that $3$ is unramified. It has to be an extension of $D3$. ```magma > I3 := InertiaField(P3); > I3; Number Field with defining polynomial $.1^4 + 80346384509631057182412*$.1^3 + 2256835583037881432653115137736209396615693022*$.\\ 1^2 + 2795818092855476469056989739955845736579291605177\\ 3809455107173769804*$.1 + 2207787685682553980385342263526644079975418801375161428\\ 41147104301325760481728833650060994 over the Rational Field > Discriminant($1); 10700005925626216180895747020647047166414333000723923591882\\ 57829873417638072117114945163507537844711544617147344227643\\ 21408503489566949866295669400825222748660907808235401444104\\ 29329493645714658394673579309893726532999745496689571082958\\ 8286937125090034449967033769822464 ``` This (polynomial) discriminant is huge, in fact it is so large that we should avoid the factorisation. We already know the discriminant of $Ka$. The discriminant of $I3$ has to be a divisor - so we can use the `Discriminant` parameter to `MaximalOrder`: (We are going to need the `MaximalOrder` for the following embedding.) ```magma > I3M := MaximalOrder(EquationOrder(I3): > Discriminant := Discriminant(Oa)); > I3M := MaximalOrder(I3); ``` $D3$ should be a subfield of $I3$, so lets verify it: ```magma > IsSubfield(D3, I3); true Mapping from: FldNum: D3 to FldNum: I3 ``` As a side-effect, Magma is now aware of the embedding and will use it. Without the `IsSubfield` call, the `RelativeField` function will fail. ```magma > I3r := RelativeField(D3, I3); > I3rM := MaximalOrder(I3r); > K3r := RelativeField(D3, Ka); > K3rM := MaximalOrder(K3r); > IsInert(K3rM!!P3 meet D3M, I3rM); true ``` The last step: verify that $P3$ is totally ramified over $I3$: ```magma > K3r := RelativeField(I3, Ka); > K3rM := MaximalOrder(K3r); > IsTotallyRamified(K3rM!!P3 meet I3M, K3rM); true ``` Using the decomposition group, we can get the splitting behaviour of any prime in any subfield of $Ka$. ```magma > L := SubgroupLattice(Gal); > [ IsNormal(Gal, L[x]) : x in [1..#L]]; [ true, true, true, true, false, false, false, false, true, true, true, true, true, true, true, false, false, false, false, true, true, true, true, true, true, true, true ] > U := L[5]; > k := FixedField(Ka, U); > kM := MaximalOrder(EquationOrder(k) : > Discriminant := Discriminant(Oa)); > kM := MaximalOrder(k); > Kr := RelativeField(k, Ka); > KrM := MaximalOrder(Kr); > P43 := Decomposition(Oa, 43)[1][1]; > V := DecompositionGroup(P43); ``` The splitting behaviour is determined by the double coset decomposition of $Gal$ with respect to $U$ and $V$: ```magma > f, I := CosetAction(Gal, U); > orbs := Orbits(f(V)); > reps := []; > for o in orbs do > _, x := IsConjugate(I, 1, Rep(o)); > Append(~reps, x @@ f); > end for; > reps; [ Id(G), (1, 2, 7, 5)(3, 8, 6, 10)(4, 12, 14, 9)(11, 16, 13, 15), (1, 7)(2, 5)(3, 6)(4, 14)(8, 10)(9, 12)(11, 13)(15, 16), (1, 8)(2, 6)(3, 5)(4, 15)(7, 10)(9, 13)(11, 12)(14, 16), ] > #reps; 4 ``` So there will be at least $4$ prime ideals over $43$ in $k$: ```magma > L := [ ]; > for i in reps do > Append(~L, kM meet KrM !! Map(i)(P43)); > end for; > [ IsPrime(x) : x in L]; [ true, true, true, true ] > LL := Decomposition(kM, 43);#LL; 4 > [ Position(L, x[1]) : x in LL]; [ 4, 3, 1, 2 ] ``` ## `FrobeniusElement(K, p): FldNum, RngIntElt -> GrpPermElt` Compute a Frobenius element at $p$ in the Galois group of the Galois closure of $K$. This is a permutation on the roots of a polynomial defining $K$, which can be recovered as `DefiningPolynomial(A)` for any Artin representation $A$ of $K$; the Frobenius element is well-defined up to conjugacy and modulo inertia. ## `Example: Nf Sig FrobeniusElement (ex-87fbb9)` We take a polynomial whose Galois group is $D_5$ and compute Frobenius elements at $p=2$ and $p=5$. They in two different conjugacy classes of 5-cycles in the Galois group. ```magma > load galpols; > f:=PolynomialWithGaloisGroup(5,2); > assert IsIsomorphic(GaloisGroup(f),DihedralGroup(5)); > K:=NumberField(f); > FrobeniusElement(K,2); (1, 5, 4, 3, 2) > FrobeniusElement(K,5); (1, 3, 5, 2, 4) ```