[#P37] Inverse Galois problem
Problem. For every finite group \(G\), there exists a finite Galois extension \(K/\mathbb{Q}\) such that \(\operatorname{Gal}(K/\mathbb{Q})\cong G\).
1Context
Classical Galois theory extracts a group from a field extension. The inverse problem prescribes the group and asks for an extension realizing it.
2Problem setup
Definition 1 (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 2 (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 1. Classical Galois theory extracts a group from a field extension. The inverse problem prescribes the group and asks for an extension realizing it.
3What 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.
1Status
Current status (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.[2][1]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
Original intake status. The cited survey describes the inverse Galois problem over Q as unsolved. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Many families of finite groups have been realized. A proposed universal argument must cover every finite group over Q.
Recorded example 1. Every finite abelian group occurs as a Galois group over Q.
Computational notes
- Explicit polynomial searches can realize particular groups without settling the universal claim.
2See also
How to cite
TheoremDB contributors, “Inverse Galois problem,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/inverse-galois-problemThis page as plain text: inverse-galois-problem.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗preprint · primary source · arXiv:1512.08708, checked 2026-07-31 · checked 2026-07-31Source 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.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.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.
- 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. ↗book · primary source · checked 2026-08-01Source use: original summary.Settles the inverse Galois problem for all finite solvable groups while leaving arbitrary finite groups over Q open.
An original CC0 restatement prepared by TheoremDB maintainers.