Karthick Babu, C.G.Mukhopadhyay, A.Sahu, S.2026-02-042024Research in Number Theory, 2024, 10, 2, pp. -https://doi.org/10.1007/s40993-024-00523-8https://idr.nitk.ac.in/handle/123456789/21102Let S={a<inf>1</inf>,a<inf>2</inf>,⋯,a<inf>n</inf>} be a finite set of non-zero integers. In [5], Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field Q(a<inf>1</inf>,a<inf>2</inf>,⋯,a<inf>n</inf>) over Q. For a positive integer d⩾3, ζ<inf>d</inf> denotes the primitive d-th root of unity. In this paper, we calculate the explicit structure of the Galois group of Q(a<inf>1</inf>,a<inf>2</inf>,⋯,a<inf>n</inf>,ζ<inf>d</inf>) over Q in terms of its action on ζ<inf>d</inf> and a<inf>i</inf> for 1⩽i⩽n. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.11L2011N1311R1111R18Cyclotomic extensionsPrimes in congruence classesQuadratic extensionsQuadratic residue