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Keywords:
Galois extension; inverse Galois problem; inflated extension; automorphism group
Galois extension; inverse Galois problem; inflated extension; automorphism group
Summary:
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 ${\bf Q}$ 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 ${\bf Q}$ has a solution for a finite group $G$, say of order $n$, then there exist algebraic number fields of degree $mn$, for any $m\ge 3$ with the same automorphism group $G$.
References:
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