Shafarevich's theorem on solvable Galois groups

In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich (1954), though Alexander Schmidt later pointed out a gap in the proof, which was fixed by Shafarevich (1989).

References

• Shafarevich, I. R. (1954), "Construction of fields of algebraic numbers with given solvable Galois group", Izv. Akad. Nauk SSSR. Ser. Mat. 18: 525–578, English translation in his collected mathematical papers 
• Shafarevich, I. R. (1989), "Factors of decreasing central series" (in Russian), Mat. Zametki 45 (3): 114–117, 128