[#P37] Inverse Galois problem

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

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Definitions and notation

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.

Remark 1. The problem asks for realization over Q, rather than over an arbitrary field.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Inverse Galois problem, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 1 Details 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Inverse Galois problem, record inverse-galois-problem-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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

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Plain text

“Inverse Galois problem.” TheoremDB. P37. Problem statement; statement text SHA-256 a28db708a7306013b292887ab732bde57ab24f65571c24dec6eb9bb100371a08. https://theoremdb.org/statement/?ref=P37

BibTeX

@misc{theoremdb-problem-a28db708a7306013b292887ab732bde57ab24f65571c24dec6eb9bb100371a08,
  title = {{Inverse Galois problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 a28db708a7306013b292887ab732bde57ab24f65571c24dec6eb9bb100371a08},
  url = {https://theoremdb.org/statement/?ref=P37}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P37] Inverse Galois problem

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3What counts as a solution

1Status

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Computational notes

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance