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.
Abelian: BoolElt Default: false
SetVerbose("AutomorphismGroup", n): Maximum: 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 [Klü97], [AK99] will be employed. If F is not a normal extension, the automorphisms are obtained by a variation of the polynomial factorisation algorithm.
Abelian: BoolElt Default: false
SetVerbose("AutomorphismGroup", n): Maximum: 3
Given an algebraic field F, that is either a simple normal extension of Q or simple abelian extension of 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 [Klü97], [AK99] will be employed. If F is not a normal extension of 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 φ from G into Aut.
> Q := RationalField();
> R<x> := PolynomialRing(Q);
> K<w> := 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:
> 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
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 Q first.
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 ∈G | s(p) = p }If F is not a normal extension of Q an error will occur.
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 ∈G | s(x) - x ∈pi + 1 for all x in M }If F is not a normal extension of Q an error will occur.
This is just an abbreviation for RamificationGroup(p, 1).
This is just an abbreviation for RamificationGroup(p, 0).
Given a normal field K over 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.
If K is not a normal extension of Q an error will occur.
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.
Given a normal field K over Q and a subfield L, compute the subgroup U of the AutomorphismGroup(K) that fixes L.This function is inverse to FixedField.
If K is not a normal extension of Q an error will occur.
Given a normal field K over 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 Q an error will occur.
Given a normal field K over Q and a number field element a, compute the subgroup U of the AutomorphismGroup(K) that fixes a.This function is inverse to FixedField.
If K is not a normal extension of Q an error will occur.
This is an abbreviation for FixedField(K, DecompositionGroup(p)) where K is the number field of the order of p.
This is an abbreviation for FixedField(K, RamificationGroup(p, i)) where K is the number field of the order of p.
This is an abbreviation for FixedField(K, RamificationGroup(p)) where K is the number field of the order of p.
This is an abbreviation for FixedField(K, InertiaField(p)) where K is the number field of the order of p.
> o := MaximalOrder(ext<Rationals()|>.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 D4 by C2.
> 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.
> P2 := Decomposition(Oa, 2)[1][1]; > P3 := Decomposition(Oa, 3)[1][1];First, the valuation of the different of Oa at P2 should be ∑i=0^∞(#G(P2, i) - 1) where G(P2, i) is the i-th ramification group.
> 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); 18According to the theory, P2 should be totally ramified over the inertia field and unramified over Q:
> 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); trueNow 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.
> D3 := DecompositionField(P3); > D3M := MaximalOrder(D3); > IsTotallySplit(3, D3M); trueThe inertia field is the maximal subfield such that 3 is unramified. It has to be an extension of D3.
> 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.)
> I3M := MaximalOrder(EquationOrder(I3): > Discriminant := Discriminant(Oa)); > I3M := MaximalOrder(I3);D3 should be a subfield of I3, so lets verify it:
> IsSubfield(D3, I3); true Mapping from: FldNum: D3 to FldNum: I3As a side-effect, Magma is now aware of the embedding and will use it. Without the IsSubfield call, the RelativeField function will fail.
> I3r := RelativeField(D3, I3); > I3rM := MaximalOrder(I3r); > K3r := RelativeField(D3, Ka); > K3rM := MaximalOrder(K3r); > IsInert(K3rM!!P3 meet D3M, I3rM); trueThe last step: verify that P3 is totally ramified over I3:
> K3r := RelativeField(I3, Ka); > K3rM := MaximalOrder(K3r); > IsTotallyRamified(K3rM!!P3 meet I3M, K3rM); trueUsing the decomposition group, we can get the splitting behaviour of any prime in any subfield of Ka.
> 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:
> 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:
> 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 ]
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.
> 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)