Fixed Field Calculator

Inputs

Provide the group order and a subgroup order that divides it.

Results appear above after you submit.

Example data table

\|G\| = [L:K]	\|H\|	[L:L^H]	[L^H:K]	Interpretation
6	2	2	3	Subgroup of size 2 gives an intermediate field of degree 3 over K.
8	4	4	2	Fixed field sits one step above K with degree 2.
12	3	3	4	Degree splits into 3 and 4 along the tower.
15	5	5	3	Works when the extension is Galois with \|G\| = 15.

These rows illustrate the degree identities; they are not claims about specific polynomials.

Formula used

For a finite Galois extension L/K with Galois group G = Gal(L/K), each subgroup H ≤ G has a fixed field:

L^H = { x ∈ L : σ(x) = x for every σ ∈ H }.

The Galois correspondence gives these degree relations:

[L : L^H] = |H|
[L^H : K] = |G| / |H|
[L : K] = [L : L^H] · [L^H : K]

How to use this calculator

Enter your base symbol K and extension symbol L (optional).
Type the group order |G|, usually the degree [L:K].
Type a subgroup order |H| that divides |G|.
Press Calculate fixed field to display degrees above the form.
Use CSV/PDF buttons to export the result table.

Fixed fields as invariants in symmetry problems

In a finite Galois extension L/K, automorphisms encode algebraic symmetry. A fixed field collects elements unchanged by every map in a chosen subgroup. This viewpoint turns field theory into a measurable structure: sizes of symmetry sets determine degrees in the tower. When you change H, you change the invariants, and the intermediate field moves accordingly.

From subgroup order to extension degree

The calculator uses the correspondence between subgroups H ≤ G and intermediate fields L^H. When |H| divides |G|, the index |G|/|H| becomes the degree [L^H:K]. Meanwhile [L:L^H] equals |H|, giving a two-step factorization of [L:K]. This is especially useful when you already know |G| from a splitting field computation.

Interpreting tower consistency checks

Practical work often starts with a known group order, such as 6 for an S3 splitting field, 8 for a dihedral-like case, or 12 for A4-related examples. Selecting H of size 2 predicts an intermediate field of degree 3 over K, while H of size 3 predicts degree 2 when |G| is 6. The product check confirms internal consistency, preventing mismatched subgroup choices or arithmetic slips in planning. It also highlights that larger subgroups correspond to smaller intermediate extensions over K.

What the calculator does not infer

Degree results describe possible layer sizes, not explicit generators. Many non-isomorphic fields can share the same degrees, and some subgroup orders may occur only for particular group structures. Normality matters: only normal subgroups correspond to intermediate fields that are themselves Galois over K. You still need polynomial data, a subgroup lattice, or computational algebra to identify the precise intermediate field inside L.

Using outputs for study and verification

Use the CSV export to record cases while studying subgroup lattices or building tables for coursework. The PDF output is convenient for homework writeups and audit trails in symbolic computations. Pair the degrees with subgroup descriptions, then compare with factorization patterns, resolvent computations, or discriminant behavior to build intuition reliably. Over time, you can spot recurring degree signatures that signal common groups, such as 6 = 2×3 or 12 = 3×4, before doing heavier algebra. These quick checks reduce errors in larger solution workflows.

FAQs

1) What is a fixed field L^H?

It is the set of elements in L that every automorphism in H leaves unchanged. It forms an intermediate field between L and K.

2) Why must |H| divide |G| here?

In a finite Galois setting, [L^H:K] equals |G|/|H|. For the degree to be an integer, |H| must divide |G|.

3) Does the calculator find an explicit polynomial for L^H?

No. It computes degree information from group and subgroup sizes. Constructing generators requires additional algebraic data or computations.

4) Can different subgroups with the same order give different fixed fields?

Yes. Distinct subgroups of the same order can correspond to different intermediate fields, even though the degree values match.

5) If |G| equals [L:K], is L/K always Galois?

Not necessarily. Equality can occur, but the key requirement is that L/K is Galois so the correspondence and degree identities apply.

6) How should I choose |H| for a target intermediate degree?

Pick |H| = |G| / d where d is your desired [L^H:K]. Then verify |H| is feasible for your group structure.