Book contents
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Galois groups through invariant relations
Published online by Cambridge University Press: 04 August 2010
- Frontmatter
- Contents
- Introduction
- Galois groups through invariant relations
- Construction of Co3. An example of the use of an integrated system for computational group theory
- Embedding some recursively presented groups
- The Dedekind-Frobenius group determinant: new life in an old problem
- Group characters and π-sharpness
- Permutation group algorithms via black box recognition algorithms
- Nonabelian tensor products of groups: the commutator connection
- Simple subalgebras of generalized Witt algebras of characteristic zero
- Applications of the Baker-Hausdorff formula in the theory of finite p-groups
- Generalizations of the restricted Burnside problem for groups with automorphisms
- The ∑m-conjecture for a class of metabelian groups
- Rings with periodic groups of units II
- Some free-by-cyclic groups
- The residually weakly primitive geometries of the Suzuki simple group Sz(8)
- Semigroup identities and Engel groups
- Groups whose elements have given orders
- The Burnside groups and small cancellation theory
- Solvable Engel groups with nilpotent normal closures
- Nilpotent injectors in finite groups
- Some groups with right Engel elements
- The growth of finite subgroups in p-groups
- Symplectic amalgams and extremal subgroups
- Primitive prime divisor elements in finite classical groups
- On the classification of generalized Hamiltonian groups
- Permutability properties of subgroups
- When Schreier transversals grow wild
- Probabilistic group theory
- Combinatorial methods: from groups to polynomial algebras
- Formal languages and the word problem for groups
- Periodic cohomology and free and proper actions on ℝn × Sm
- On modules over group rings of soluble groups of finite rank
- On some series of normal subgroups of the Gupta-Sidki 3-group
Summary
Prolegomena
Let f ∈ ℚ[x] be an irreducible polynomial of degree n. Then the splitting field L ≥ ℚ of f is a normal extension. We want to determine the Galois group G = Gal(f) = Gal(L/ℚ) of f which is the group of all field automorphisms of this extension. This task is basic in computational number theory [Coh93] as the Galois group determines a lot of properties of the field extension defined by f.
Because the index [L : ℚ] might be large, however, we do not intend to construct L and thus cannot give explicitly the automorphism action of G on L. Instead we consider the action of G on the roots {α1, …, αn) of f. As f is defined over the rationals the set of these root must remain invariant under G. This permutation action is faithful, because L can be obtained by adjoining all the αi to ℚ. This action has to be transitive because f is irreducible. In other words: For a fixed arrangement of the roots, G can be considered as a transitive subgroup of Sn. We will utilize this embedding without explicitly mentioning it.
While the problem itself is initially number-theoretic, the approaches to solve it are mainly based on commutative algebra and permutation group theory. This paper presents a new approach, approximating the Galois group by its closures (subgroups of Sn that stabilizer orbits of G).
- Type
- Chapter
- Information
- Groups St Andrews 1997 in Bath , pp. 379 - 393Publisher: Cambridge University PressPrint publication year: 1999
- 3
- Cited by