This chapter deals with Galois groups and automorphism of number fields and several kinds of function fields. It also deals with computing the subfields of these kinds of fields. While these problems are closely related from a theoretical point of view (basically, everything is determined by the Galois group), as algorithmic problems they are quite different.

The first task, that of computing automorphisms of normal extensions of Q (and of abelian extensions of number fields) can be thought of a special case of factorisation of polynomials over number fields: the automorphisms of a number field are in one-to-one correspondence with the roots of the defining equation in the field. However, the computation follows a different approach and is based on some combinatorial properties. It should be noted, though, that the algorithms only apply to normal fields; i.e., they cannot be used to find non-trivial automorphisms of non-normal fields!

The second task, namely that of computing the Galois group of the normal closure of a number field, is of course closely related to the problem of computing the Galois group of a polynomial. The method implemented in Magma allows the computation of Galois groups of polynomials (and number fields) of arbitrarily high degrees and is independent of the classification of transitive permutation groups. The result of the computation of a Galois group will be a permutation group acting on the roots of the (defining) polynomial, where the roots (or approximations of them) are explicitly computed in some suitable p-adic field; thus the splitting field is not (directly) part of the computation. The explicit action on the roots allows one, for example, to compute algebraic representations of arbitrary subfields of the splitting field, even the splitting field itself, provided the degree is not too large.

The last main task dealt with in this chapter is the computation of subfields of a number field. While of course this can be done using the main theorem of Galois theory (the correspondence between subgroups and subfields), the computation is completely independent; in fact, the computation of subfields is usually the first step in the computation of the Galois group. The algorithm used here is mainly combinatorical.

Finally, this chapter also deals with applications of the Galois theory:

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the computation of subfields and subfield towers of the splitting field

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solvability by radicals: if the Galois group of a polynomial is solvable, the roots of the polynomial can be represented by (iterated) radicals.

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basic Galois-cohomology; i.e., the action of the automorphisms on the ideal class group, the multiplicative group of the field and derived objects.

 
Automorphism Groups
      Automorphisms(F) : FldAlg -> [ Map ]
      AutomorphismGroup(F) : FldAlg -> GrpPerm, PowMap, Map
      Example RngOrdGal_Automorphisms (H40E1)
      AutomorphismGroup(K, F) : FldAlg, FldAlg -> GrpPerm, PowMap, Map
      DecompositionGroup(p) : RngIntElt -> GrpPerm
      RamificationGroup(p, i) : RngOrdIdl, RngIntElt -> GrpPerm
      RamificationGroup(p) : RngOrdIdl -> GrpPerm
      InertiaGroup(p) : RngOrdIdl -> GrpPerm
      FixedField(K, U) : FldAlg, GrpPerm -> FldNum, Map
      FixedField(K, S) : FldAlg, [Map] -> FldAlg, Map
      FixedGroup(K, L) : FldAlg, FldAlg -> GrpPerm
      FixedGroup(K, L) : FldAlg, [FldAlgElt] -> GrpPerm
      FixedGroup(K, a) : FldAlg, FldAlgElt -> GrpPerm
      DecompositionField(p) : RngOrdIdl -> FldNum, Map
      RamificationField(p, i) : RngOrdIdl, RngIntElt -> FldNum, Map
      RamificationField(p) : RngOrdIdl -> FldNum, Map
      InertiaField(p) : RngOrdIdl -> FldNum, Map
      Example RngOrdGal_Ramification (H40E2)
      FrobeniusElement(K, p) : FldNum, RngIntElt -> GrpPermElt
      Example RngOrdGal_nf-sig-FrobeniusElement (H40E3)

 
Galois Groups
      GaloisGroup(f) : RngUPolElt[RngInt] -> GrpPerm, SeqEnum, GaloisData
      IsEasySnAn(f) : RngUPolElt -> RngIntElt
      GaloisGroup(K) : FldNum -> GrpPerm, SeqEnum, GaloisData
      GaloisProof(f, S) : RngUPolElt, GaloisData -> BoolElt
      GaloisRoot(f, i, S) : RngUPolElt, RngIntElt, GaloisData -> RngElt
      GaloisGroupConjugation(S, T) : GaloisData, GaloisData -> GrpPermElt
      GaloisAutomorphismGroup(K) : FldNum[FldRat] -> GrpPerm, Map
      Stauduhar(G, H, S, B) : GrpPerm, GrpPerm, GaloisData, RngIntElt -> RngIntElt, GrpPermElt, BoolElt, RngSLPolElt
      IsInt(x, B, S) : RngElt, RngIntElt, GaloisData -> BoolElt, RngElt
      Example RngOrdGal_GaloisGroups (H40E4)

      Straight-line Polynomials
            SLPolynomialRing(R, n) : Rng, RngIntElt -> RngSLPol
            Name(R, i) : RngSLPol, RngIntElt -> RngSLPolElt
            ElementarySymmetricPolynomial(R, i) : RngSLPol, RngIntElt -> RngSLPolElt
            BaseRing(R) : RngSLPol -> Rng
            Rank(R) : RngSLPol -> RngIntElt
            SetEvaluationComparison(R, F, n) : RngSLPol, FldFin, RngIntElt ->
            GetEvaluationComparison(R) : RngSLPol -> FldFin, RngIntElt
            InitializeEvaluation(R, S) : RngSLPol, [RngElt] ->
            Derivative(x, i) : RngSLPolElt, RngIntElt -> RngSLPolElt
            Evaluate(f) : RngSLPolElt -> RngElt

      Invariants
            GaloisGroupInvariant(G, H) : GrpPerm, GrpPerm -> RngSLPolElt
            RelativeInvariant(G, H) : GrpPerm, GrpPerm -> RngSLPolElt
            CombineInvariants(R, G, H1, H2, H3) : Rng, GrpPerm, Tup<GrpPerm, RngSLPolElt>, Tup<GrpPerm, RngSLPolElt>, GrpPerm -> RngSLPolElt
            IsInvariant(F, p) : RngSLPolElt, GrpPermElt -> BoolElt
            Bound(I, B) : RngSLPolElt, RngIntElt -> RngIntElt
            Bound(I, B) : RngSLPolElt, RngElt -> RngElt
            PrettyPrintInvariant(I) : RngSLPolElt -> MonStgElt

      Subfields and Subfield Towers
            GaloisSubgroup(K, U) : FldNum, GrpPerm -> RngUPolElt, RngSLPolElt
            GaloisQuotient(K, Q) : FldNum, GrpPerm -> SeqEnum[RngUPolElt]
            GaloisSubfieldTower(S, L) : GaloisData, [GrpPerm] -> FldNum, [Tup<RngSLPolElt, RngUPolElt, [GrpPermElt]>], UserProgram, UserProgram
            GaloisSplittingField(f) : RngUPolElt -> FldNum, [FldNumElt], GrpPerm, [[FldNumElt]]
            Example RngOrdGal_galois-subfield (H40E5)

      Solvability by Radicals
            SolveByRadicals(f) : RngUPolElt -> FldNum, [FldNumElt], [FldNumElt]
            CyclicToRadical(K, a, z) : FldNum, FldNumElt, RngElt -> FldNum, [FldNumElt], [FldNumElt]
            Example RngOrdGal_solve-radical (H40E6)

      Linear Relations
            LinearRelations(f) : RngUPolElt -> Mtrx, GaloisData
            LinearRelations(f, I) : RngUPolElt, [RngSLPolElt] -> Mtrx, GaloisData
            VerifyRelation(f, F) : RngUPolElt, RngSLPolElt -> BoolElt
            Example RngOrdGal_linear-relations (H40E7)

      Other
            ConjugatesToPowerSums(I) : [] -> []
            PowerSumToElementarySymmetric(I) : [] -> []

 
Subfields
      Subfields(K, n) : FldAlg, RngIntElt -> [ < FldAlg, Hom > ]
      Subfields(K) : FldAlg -> [ < FldAlg, Hom > ]

      The Subfield Lattice
            SubfieldLattice(K) : FldNum -> SubFldLat
            # L : SubFldLat -> RngIntElt
            Bottom(L) : SubFldLat -> SubFldLatElt
            Top(L) : SubFldLat -> SubFldLatElt
            Random(L) : SubFldLat -> SubFldLatElt
            L ! n : SubFldLat, RngIntElt -> SubFldLatElt
            NumberField(e) : SubFldLatElt -> FldNum
            EmbeddingMap(e) : SubFldLatElt -> Map
            Degree(e) : SubFldLatElt -> RngIntElt
            e eq f : SubFldLatElt, SubFldLatElt -> BoolElt
            e subset f : SubFldLatElt, SubFldLatElt -> BoolElt
            e * f : SubFldLatElt, SubFldLatElt -> SubFldLatElt
            e meet f : SubFldLatElt, SubFldLatElt -> SubFldLatElt
            &meet S : [ SubFldLatElt ] -> SubFldLatElt
            MaximalSubfields(e) : SubFldLatElt -> [ SubFldLatElt ]
            MinimalOverfields(e) : SubFldLatElt -> [ SubFldLatElt ]
            Example RngOrdGal_SubfieldLattice (H40E8)

 
Galois Cohomology
      Hilbert90(a, M) : FldNumElt, Map[FldNum, FldNum] -> FldNumElt
      SUnitCohomologyProcess(S, U) : {RngOrdIdl}, GrpPerm -> {1}
      IsGloballySplit(C, l) : , UserProgram -> BoolElt, UserProgram
      IsSplitAsIdealAt(I, l) : RngOrdFracIdl, UserProgram -> BoolElt, UserProgram, [RngOrdIdl]

 
Automorphisms of Fields

      Creation of Field Automorphisms
            FieldAutomorphism(F, g) : Fld, GrpPermElt -> FldAut
            FieldAutomorphism(F, a) : Fld, Map[Fld,Fld] -> FldAut
            IdentityAutomorphism(F) : Fld -> FldAut
            FieldAutomorphism(K) : AlgAss[Fld] -> FldAut
            ChangeRing(a, K) : FldAut, Fld -> FldAut

      Properties of Field Automorphisms
            BaseField(a) : FldAut -> Fld
            Order(a) : FldAut -> RngIntElt
            FixedField(a) : FldAut -> Fld
            Automorphism(a) : FldAut -> Map[Fld, Fld]
            FixedFieldSymbol(a, P) : FldAut, RngOrdIdl -> RngIntElt

      Predicates on Field Automorphisms
            IsIdentity(a) : FldAut -> BoolElt

      Arithmetic of Field Automorphisms
            a * b : FldAut, FldAut -> FldAut
            a ^ n : FldAut, RngIntElt -> FldAut
            Inverse(a) : FldAut -> FldAut
            a eq b : FldAut, FldAut -> BoolElt
            x @ a : FldElt, FldAut -> FldElt
            v @ a : ModTupFldElt[Fld], FldAut -> ModTupFldElt[Fld]
            M @ a : ModMatFldElt[Fld], FldAut -> ModMatFldElt[Fld]
            I @ a : RngOrdFracIdl[FldOrd], FldAut -> RngOrdFracIdl[FldOrd]
            Trace(P, a) : RngOrdFracIdl, FldAut -> RngOrdFracIdl
            Norm(P, a) : RngOrdFracIdl, FldAut -> RngOrdFracIdl

 
Bibliography

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