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THE INVERSE GALOIS PROBLEM

  • 투고 : 2021.10.21
  • 심사 : 2022.02.24
  • 발행 : 2022.05.30

초록

The inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers. This problem, first posed in the early 19th century, is unsolved. In other words, we consider a pair - the group G and the field K. The question is whether there is an extension field L of K such that G is the Galois group of L. In this paper we present the proof that any group G is a Galois group of any field extension. In other words, we only consider the group G. And we present the solution to the inverse Galois problem.

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참고문헌

  1. I.R. Shafarevich, Construction of fields of algebraic numbers with given solvable Galois group, Izv. Akad. Nauk SSSR. Ser. Mat. 18 (1954), 525-578.
  2. M. Dugas, R. Gobel, All infinite groups are Galois groups over any field, Transactions of the American Mathematical Society, 304 (1987).
  3. I.R. Shafarevich, Factors of decreasing central series, Mat. Zametki (in Russian) 45 (1987), 114-117.
  4. H.G.J. Tiesinga, The inverse Galois problem, Bachelor Project Mathematics, University of Groningen, 2016.