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<p>In 2018, Legrand and Paran proved a weaker form of the inverse Galois problem: Every finite group appears as the automorphism group of infinitely many finite (possibly non-Galois) extensions of a given Hilbertian base field. For <b>Q</b> it was proved earlier by Fried. Our objective is to determine how big the degree of such extension can be when compared to the order of the automorphism group. A special case of our result shows that if the inverse Galois problem for <b>Q</b> has a solution for a finite group <i>G</i>, say of order <i>n</i>, then there exist algebraic number fields of degree <i>mn</i>, for any <i>m</i> ⩾ 3 with the same automorphism group <i>G</i>.</p>

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Inflated G-extensions for algebraic number fields

  • 期刊论文

  • 发表日期：

    2026年4月24日

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  • M. Krithika,
  • Pichai Vanchinathan

摘要

In 2018, Legrand and Paran proved a weaker form of the inverse Galois problem: Every finite group appears as the automorphism group of infinitely many finite (possibly non-Galois) extensions of a given Hilbertian base field. For Q it was proved earlier by Fried. Our objective is to determine how big the degree of such extension can be when compared to the order of the automorphism group. A special case of our result shows that if the inverse Galois problem for Q has a solution for a finite group G, say of order n, then there exist algebraic number fields of degree mn, for any m ⩾ 3 with the same automorphism group G.