For a field K and a finite group G, say that G is realizable over K if there exists a Galois extension L/K with operatornameGal(L/K) cong G. Every finite group G is realizable over the field Q of rational numbers: there exists a finite Galois extension K/Q, i.e. a field K with Q ⊆ K, [K:Q] < ∞, and K normal and separable over Q, such that operatornameGal(K/Q) = operatornameAut(K/Q) cong G. Equivalently, for every finite group G there is a monic irreducible polynomial f ∈ Q[x] whose splitting field K over Q satisfies operatornameGal(K/Q) cong G.

inverse Galois problem

At least 133 years old · documented by 1892

For a field $K$ and a finite group $G$ , say that $G$ is realizable over $K$ if there exists a Galois extension $L/K$ with $\operatorname{Gal}(L/K) \cong G$ .

Every finite group $G$ is realizable over the field $\mathbb{Q}$ of rational numbers: there exists a finite Galois extension $K/\mathbb{Q}$ , i.e. a field $K$ with $\mathbb{Q} \subseteq K$ , $[K:\mathbb{Q}] < \infty$ , and $K$ normal and separable over $\mathbb{Q}$ , such that

\operatorname{Gal}(K/\mathbb{Q}) = \operatorname{Aut}(K/\mathbb{Q}) \cong G.

Equivalently, for every finite group $G$ there is a monic irreducible polynomial $f \in \mathbb{Q}[x]$ whose splitting field $K$ over $\mathbb{Q}$ satisfies $\operatorname{Gal}(K/\mathbb{Q}) \cong G$ .

References

Primary source

Wikipedia

Additional references

Wikipedia, Inverse Galois problem, the article this problem comes from.

group-theory unsolved-problem

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed progress

The general question remains open, while August claims cover one long-standing exceptional case and all groups in degree 24, without independent verification.

The inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of $\mathbb{Q}$ . Posed in the early nineteenth century, it remains unresolved in general.

Known results

Hilbert realized $S_n$ and $A_n$ over $\mathbb{Q}$ .
Scholz and Reichardt (1937) realized every odd $p$ -group.
Shafarevich (1954; proof corrected in 1989) realized every finite solvable group.
By 1989, $25$ of the $26$ sporadic simple groups were realized over $\mathbb{Q}$ .

August 2026 claimed advances

On August 11, 2026, Huang, Jackson, Lee, Poonen, Pries, and Zhang were reported to have constructed a degree- $23$ polynomial with Galois group $M_{23}$ , completing the sporadic cases; the claim is unverified. On August 8, 2026, the IGP24 competition announced realizations for all approximately $25{,}000$ degree- $24$ groups, also without independent verification. Neither claim resolves arbitrary finite groups.

Current status (as of September 2026): Realizability over $\mathbb{Q}$ is settled for all finite solvable groups and many nonsolvable families; recent claims cover $M_{23}$ and degree- $24$ groups, but the general problem remains open and those claims are unverified.

Sources

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