Contents

Intermediate FieldEdit

An intermediate field is a field K that lies between two other fields in an extension: F ⊆ K ⊆ E. In the language of abstract algebra, K is one of the subfields that sit inside an overfield E and above a base field F. The study of intermediate fields is central to understanding how a field extension E/F is built from simpler pieces, and it provides a rigorous way to organize the algebraic structure of the roots of polynomials and the symmetries that act on them. This perspective is especially productive when E/F is finite or when the extension carries a rich symmetry group.

In the standard setting, one considers field extensions E over F and examines the subfields that satisfy F ⊆ K ⊆ E. The lattice of such intermediate fields — organized by inclusion — encodes how complex algebraic elements in E can be constructed from elements algebraic over F. The subject becomes particularly powerful when paired with group theory through the Galois correspondence, which links subfields to subgroups of the automorphism group of E that fixes F.

This article surveys the notion of an intermediate field, emphasizes the classical results that connect subfields to symmetry groups, and sketches typical examples and extensions where the correspondence is most transparent. For readers encountering this concept in the study of polynomials, number theory, or algebraic geometry, intermediate fields serve as a pivotal bridge between concrete elements and abstract structure.

Basic definitions

  • intermediate field: A field K with F ⊆ K ⊆ E, written as F ⊆ K ⊆ E, where E is an overfield of F. See field extension for the broader framework of how E is built from F.
  • field extension: The inclusion E ⊇ F together with the structure of E as a field containing F; often denoted as E/F. See field extension for formal definitions and examples.
  • Galois extension: A field extension E/F that is normal and separable; this setting is where the most powerful correspondence between subfields and subgroups lives. See Galois theory and Galois group.
  • lattice of intermediate fields: The partially ordered set of all intermediate fields between F and E, ordered by inclusion. It forms a lattice that reflects how intermediate fields fit together.
  • fixed field: Given a group H of automorphisms of E that fix F, the fixed field E^H consists of all elements of E that are unchanged by every element of H. See fixed field.

Examples help illustrate the notion. If F = Q and E = F(√2, √3) (the field obtained by adjoining √2 and √3 to Q), the intermediate fields include Q(√2), Q(√3), Q(√6), and Q itself. Each of these sits between Q and E, and their arrangement forms a lattice reflecting the algebraic relations among √2 and √3. The symmetry group that stabilizes Q inside E is the Galois group, which in this case has a small and explicit structure linked to these subfields.

The Galois correspondence

In the finite, well-behaved case, there is a deep and precise relationship between subfields and symmetry groups:

  • finite Galois extensions: If E/F is a finite Galois extension with Galois group Gal(E/F), there is a one-to-one, inclusion-reversing correspondence between subgroups H ≤ Gal(E/F) and intermediate fields K with F ⊆ K ⊆ E. The correspondence is given by K = E^H and H = Gal(E/K). See Galois theory and Galois group.
  • fixed-field construction: For a subgroup H ≤ Gal(E/F), the fixed field E^H is the set of elements of E fixed by all automorphisms in H. This operation ties subgroups directly to intermediate fields. See fixed field.

This correspondence explains why certain polynomials over F have solvable or unsolvable radical expressions: it ties solvability to the structure of the Galois group and, through it, to the lattice of subfields. Classic examples include the field E = Q(√[n]{a}) over F = Q when a is chosen to yield a nontrivial Galois group; the intermediate fields reflect the subgroups of the automorphism group that permute the roots of the minimal polynomial.

Examples and computations

  • E = Q(√2, √3) over F = Q has a Galois group isomorphic to the Klein four-group, and the intermediate fields are precisely the fields generated by each subset of the square roots, together with Q and E. The intermediate fields include Q(√2), Q(√3), Q(√6), as well as E and Q.
  • In a non-Galois or infinite extension, the correspondence becomes more nuanced: not every subfield necessarily corresponds to a normal subgroup, and the lattice of intermediate fields can be richer or more intricate. See Galois theory for the scope and limits of the correspondence.

Infinite and non-Galois contexts

In infinite extensions, intermediate fields can form a more complicated lattice, and the direct one-to-one correspondence with subgroups of a finite automorphism group no longer holds in the same form. Nevertheless, many of the same ideas persist: fixed fields of appropriate groups continue to provide a way to organize subfields, and the study of automorphisms of E that fix F remains a central tool. See field extension and Galois theory for discussions of these generalizations.

Applications and connections

Intermediate fields appear across several areas of mathematics:

  • Number theory: intermediate fields of number field extensions reveal how primes split and how units and class groups interact inside larger fields. See number theory and related discussions of algebraic number theory.
  • Algebraic geometry: function fields of varieties often involve intermediate fields when studying rational maps and birational equivalence.
  • Algebraic solvability: the Galois correspondence provides a clear criterion for when a polynomial can be solved by radicals, linking the existence of certain intermediate fields to the structure of the Galois group.
  • Cryptography: some cryptographic constructions rely on the arithmetic of field extensions and subfields, where understanding the lattice of intermediate fields informs security and efficiency considerations. See cryptography for broader context.

See also