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GALOIS THEORY OF NUMBER FIELDS

The Galois theory of number fields deals with the group of automorphisms of a number field, the group of automorphisms of the normal closure of a number field and with the subfields of a given number field. While all three problems are, at least in theory, dealt with easily using the main theorems of Galois theory, they correspond to completely different and independent algorithmic problems.

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.
 
Acknowledgements
 
Automorphism Groups
 
Galois Groups
      Straight-line Polynomials
      Invariants
      Subfields and Subfield Towers
      Solvability by Radicals
      Linear Relations
      Other
 
Subfields
      The Subfield Lattice
 
Galois Cohomology
 
Bibliography







DETAILS

 
Automorphism Groups
      Automorphisms(F) : FldAlg -> [ Map ]
      AutomorphismGroup(F) : FldAlg -> GrpPerm, PowMap, Map
      Example RngOrdGal_Automorphisms (H38E1)
      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] -> FldNum, 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 (H38E2)
      FrobeniusElement(K, p) : FldNum, RngIntElt -> GrpPermElt
      Example RngOrdGal_nf-sig-FrobeniusElement (H38E3)

 
Galois Groups
      GaloisGroup(f) : RngUPolElt[RngInt] -> GrpPerm, SeqEnum, GaloisData
      GaloisGroup(K) : FldNum -> GrpPerm, SeqEnum, GaloisData
      GaloisProof(f, S) : RngUPolElt, GaloisData -> BoolElt
      GaloisRoot(f, i, S) : RngUPolElt, RngIntElt, GaloisData -> RngElt
      GaloisRoot(i, S) : RngIntElt, GaloisData -> RngElt
      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 (H38E4)

      Straight-line Polynomials
            SLPolynomialRing(R, n) : Rng, RngIntElt -> RngSLPol
            Name(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
            Derivative(x, i) : RngSLPolElt, RngIntElt -> RngSLPolElt

      Invariants
            GaloisGroupInvariant(G, H) : GrpPerm, GrpPerm -> RngSLPolElt
            RelativeInvariant(G, H) : GrpPerm, GrpPerm -> RngSLPolElt
            CombineInvariants(G, H1, H2, H3) : 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

      Subfields and Subfield Towers
            GaloisSubgroup(K, U) : FldNum, GrpPerm -> FldNum, UserProgram
            GaloisQuotient(K, Q) : FldNum, GrpPerm -> SeqEnum[FldNum]
            GaloisSubfieldTower(S, L) : GaloisData, [GrpPerm] -> FldNum, [Tup<RngSLPolElt, RngUPolElt, [GrpPermElt]>], UserProgram, UserProgram
            GaloisSplittingField(f) : RngUPolElt -> FldNum, [FldNumElt], GrpPerm, [[FldNumElt]]
            Example RngOrdGal_galois-subfield (H38E5)

      Solvability by Radicals
            SolveByRadicals(f) : RngUPolElt -> FldNum, [FldNumElt], [FldNumElt]
            CyclicToRadical(K, a, z) : FldNum, FldNumElt, RngElt -> FldNum, [FldNumElt], [FldNumElt]
            Example RngOrdGal_solve-radical (H38E6)

      Linear Relations
            LinearRelations(f) : RngUPolElt -> Mtrx, GaloisData
            LinearRelations(f, I) : RngUPolElt, [RngSLPolElt] -> Mtrx, GaloisData
            VerifyRelation(f, F) : RngUPolElt, RngSLPolElt -> BoolElt
            Example RngOrdGal_linear-relations (H38E7)

      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 (H38E8)

 
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]

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013