Realizable classes of tetrahedral extensions Original Research Article

Let k be a number field and Ok its ring of integers. Let Γ be the alternating group A4. Let image be a maximal Ok-order in k[Γ] containing Ok[Γ] and image its class group. We denote by image the set of realizable classes, that is the set of classes image such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of image is equal to c, where ON is the ring of integers of N. In this article we determine image and we prove that it is a subgroup of image provided that k and the 3rd cyclotomic field of image are linearly disjoint, and the class number of k is odd.