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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-on correspondence to 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 applies to normal
fields, ie. they cannot be usd 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 arbitrary high degrees and is independent
on 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
computations. The explicit action on the roots allows one to
for example 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 deasl with applications of the Galois theory:
- -
- the computation of subfields and subfield towers of the splitting
field
- -
- solvability by radicals: in the Galois group of a polynomial
is solvable, the roos of the polynomial can be represented by
(iterated) radicals.
- -
- basic Galois-cohomology, ie. the action of the automorphisms
on the ideal class group, the multiplicative group of the field
and derived objects.
Acknowledgements
Automorphism Groups
Galois Groups
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 (H34E1)
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 (H34E2)
Galois Groups
GaloisGroup(L) : FldAlg[FldAlg] -> GrpPerm, [ FldPrElt ], Any
GaloisGroup(f) : RngUPolElt[RngIntElt] -> GrpPerm, SeqEnum, GaloisData
GaloisGroup(K) : FldAlg[FldRat] -> 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, UserProgram
IsInt(x, B, S) : RngElt, RngIntElt, GaloisData -> BoolElt, RngElt
Example RngOrdGal_GaloisGroups (H34E3)
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
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 (H34E4)
Solvability by Radicals
SolveByRadicals(f) : RngUPolElt -> FldNum, [FldNumElt]
CyclicToRadical(K, a, z) : FldNum, FldNumElt, RngElt -> FldNum, [FldNumElt], [FldNumElt]
Example RngOrdGal_solve-radical (H34E5)
Linear Relations
LinearRelations(f) : RngUPolElt -> Mtrx, GaloisData
LinearRelations(f, I) : RngUPolElt, [RngSLPolElt] -> Mtrx, GaloisData
VerifyRelation(f, F) : RngUPolElt, RngSLPolElt -> BoolElt
Example RngOrdGal_linear-relations (H34E6)
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 (H34E7)
Galois Cohomology
Hilbert90(a, M) : FldNumElt, Map[FldNum, FldNum] -> FldNumElt
SUnitCohomologyProcess(S, U) : {RngOrdIdl}, GrpPerm -> {1}
IsGloballySplit(C, l) : , UserProgram -> BoolElt, UserProgram
Bibliography
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