Calculator Inputs
Example data table
A common teaching example uses an order-6 group. Subgroup orders 1,2,3,6 give indices 6,3,2,1 and corresponding fixed-field degrees.
| Example subgroup | |H| | [G:H] | Fixed field | [EH:F] | [E:EH] | Normal? |
|---|---|---|---|---|---|---|
| {e} (trivial) | 1 | 6 | E^{e} = E | 6 | 1 | Yes |
| C2 (a transposition subgroup) | 2 | 3 | E^{C2} | 3 | 2 | Usually no |
| A3 (order-3 subgroup) | 3 | 2 | E^{A3} | 2 | 3 | Yes |
| G (whole group) | 6 | 1 | E^{G} = F | 1 | 6 | Yes |
Professional article
Correspondence objectives
This tool supports coursework and review by translating subgroup data into field-degree implications for a finite Galois extension. You provide the group order and a list of candidate subgroups, then the interface computes indices, fixed-field degrees, and quick validity checks that help you reason about intermediate fields. A built‑in order‑6 example mirrors common S3 lectures and lets students compare multiple subgroup sizes at once.
Inputs and assumptions
Because the full group structure is not derived automatically, the calculator treats each subgroup as a named record with an order and an optional normality flag. The central assumption is the standard Galois correspondence between subgroups of the Galois group and intermediate fields of the extension, with inclusion reversing. If you enter a degree different from |G|, the output also surfaces a consistency note so you can revisit definitions.
Degree and index outputs
For each subgroup H, the key computation is the index [G:H] when |H| divides |G|. The tool reports [E^H:F]=[G:H] and [E:E^H]=|H|, giving a consistent degree budget that can be compared across multiple subgroups to spot impossible or inconsistent entries. Rows whose orders do not divide |G| are flagged, which is useful when you are testing conjectured subgroup lists from scratch work.
Normality and quotients
Normality matters when you want to identify normal intermediate extensions and quotient Galois groups. If H is marked normal, the tool highlights that E^H/F is normal and that Gal(E^H/F) is isomorphic to G/H in the classical setting, supporting fast reasoning about solvability and tower structure. In practice, this helps you separate “field degree” calculations from “field symmetry” questions when organizing solutions.
Using exports in practice
Export features turn computed tables into study assets. CSV is convenient for sorting by subgroup order, while the PDF export produces a portable summary for notes and teaching. By iterating with different subgroup lists, you can map candidate lattices, verify degree constraints, and document conclusions for problem sets. Many users keep one export per exercise, so later corrections remain traceable and easy to audit. It fits seminars, tutorials, and quick reviews.
FAQs
It computes the index [G:H]=|G|/|H| when divisible, then reports [E^H:F]=[G:H] and [E:E^H]=|H| for each listed subgroup.
A “Check” row means the entered subgroup order does not divide |G|, so the index is not an integer. Reconfirm the subgroup order or the group order before interpreting degree relations.
No. You supply subgroup names and orders. The calculator focuses on degree bookkeeping and correspondence logic, not group enumeration or computational algebra.
Mark Normal when H is normal in G. The output then reminds you that E^H/F is normal and that the quotient G/H is the relevant Galois group for the intermediate extension.
In a finite Galois extension, these typically match. If you enter different values, the tool shows a note and still computes indices using |G| so subgroup-to-field degrees remain consistent.
CSV is ideal for sorting and filtering subgroup data in spreadsheets. PDF creates a clean, shareable snapshot for notes, grading, or study groups without re-running the calculation.