# P37: Inverse Galois problem

- ID: `P37`
- Reference: `inverse-galois-problem`
- Page: https://theoremdb.org/statements/P37
- Record maturity: Reviewed problem with recorded work

## Problem

For every finite group \(G\), there exists a finite Galois extension \(K/\mathbb{Q}\) such that \(\operatorname{Gal}(K/\mathbb{Q})\cong G\).

### Context

Classical Galois theory extracts a group from a field extension. The inverse problem prescribes the group and asks for an extension realizing it.

### Problem setup

- **Definition (A finite Galois extension L over Q).** A finite Galois extension L over Q is a finite field extension whose automorphisms fixing Q form its Galois group.
- **Definition (The problem asks for realization over Q, rather than over an arbitrary field).** The problem asks for realization over Q, rather than over an arbitrary field.
- **Remark.** Classical Galois theory extracts a group from a field extension. The inverse problem prescribes the group and asks for an extension realizing it.

### What counts as a solution

- Construct or prove the existence of a finite Galois extension of Q for every finite group, or exhibit a finite group and prove that it cannot occur as such a Galois group.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: All finite solvable groups and many nonsolvable families are known to occur over Q, but the linked survey describes realization of every finite group over Q as unsolved. Exact unresolved remainder: Construct or prove a finite Galois extension of Q for every finite group, or exhibit a finite group and prove it cannot occur as such a Galois group. [2](#reference-2) [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: All finite solvable groups and many nonsolvable families are known to occur over Q, but the linked survey describes realization of every finite group over Q as unsolved. Exact unresolved remainder: Construct or prove a finite Galois extension of Q for every finite group, or exhibit a finite group and prove it cannot occur as such a Galois group.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: All finite solvable groups and many nonsolvable families are known to occur over Q, but the linked survey describes realization of every finite group over Q as unsolved.

Exact unresolved remainder: Construct or prove a finite Galois extension of Q for every finite group, or exhibit a finite group and prove it cannot occur as such a Galois group.

### Background and intake notes

- Original intake status: The cited survey describes the inverse Galois problem over Q as unsolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited survey and current literature on 2026-07-22.
- Many families of finite groups have been realized. A proposed universal argument must cover every finite group over Q.

- Recorded example: Every finite abelian group occurs as a Galois group over Q.

### Open directions

- **Route 1** (reported): Construct or prove the existence of a finite Galois extension of Q for every finite group, or exhibit a finite group and prove that it cannot occur as such a Galois group. [1](#reference-1)

### Computational notes

- Explicit polynomial searches can realize particular groups without settling the universal claim.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `inverse-galois-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Fariba Ranjbar and Saeed Ranjbar, “Inverse Galois Problem and Significant Methods”. arXiv:1512.08708 (2015). Fariba Ranjbar and Saeed Ranjbar, arXiv:1512.08708, abstract and survey https://arxiv.org/abs/1512.08708
   - Also cited at abstract and survey of Hilbert irreducibility, Noether, rigidity, and known group families
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:1512.08708, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited survey describes the inverse Galois problem over Q as unsolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Summarizes major positive classes and states the universal problem over Q as open.
   - Source named by the research packet.
2. <a id="reference-2"></a>I. R. Šafarevič, “Construction of fields of algebraic numbers with given solvable Galois group”. American Mathematical Society Translations: Series 2 (1956), 185-237. DOI 10.1090/trans2/004/08. main theorem constructing a Galois extension of Q for every finite solvable group https://doi.org/10.1090/trans2/004/08
   - book; primary source; checked 2026-08-01
   - Source use: original_summary
   - Settles the inverse Galois problem for all finite solvable groups while leaving arbitrary finite groups over Q open.