“…Gurak [Gur78a] showed that the Hasse norm principle holds for Galois extension K/k if all the Sylow subgroups of Gal(K/k) are cyclic. Note that this also follows from Theorem 3.6 and the retract k-rationality of T = R For non-Galois extension K/k of degree n, the Hasse norm principle was investigated by Bartels [Bar81a] (holds for n = p; prime), [Bar81b] (holds for G ≃ D n ), Voskresenskii and Kunyavskii [VK84] (holds for G ≃ S n ), Kunyavskii [Kun84] (n = 4), Drakokhrust and Platonov [DP87] (n = 6), Macedo [Mac20] (holds for G ≃ A n (n = 4)), Macedo and Newton [MN22] (G ≃ A 4 , S 4 , A 5 , S 5 , A 6 , A 7 (general n)), Hoshi, Kanai and Yamasaki [HKY22] (n ≤ 15 (n = 12)) (holds for G ≃ M n (n = 11, 12, 22, 23, 24; 5 Mathieu groups)), [HKY23] (n = 12), [HKY] (G ≃ M 11 , J 1 (general n)), Hoshi and Yamasaki [HY2] (holds for G ≃ PSL 2 ( 7 ) (n = 21), PSL 2 ( 8 ) (n = 63)) where G = Gal(L/k) and L/k is the Galois closure of K/k. Recall that the case where n = p also follows from Theorem 3.6 and the retract k-rationality of T = R Let T = R…”