The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field K that is Galois over Q or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of K, for the quotient E_K of the unit group of the ring of integers of K modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real p-rational number fields, we prove that the non-abelian p-rational p-extensions of Q do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of E_L for varying Galois extensions L/F of totally real p-rational number fields whose Galois groups are isomorphic to a fixed finite group G. In that case, we prove that there exists a finite set Ω of Z_p[G]-lattices such that for every L, Z_p ⊗_Z E_L is factor equivalent to Z_p[G]ⁿ ⊕ X as Z_p[G]-lattices for some X in Ω and a non-negative integer n.

We would like to thank David Burns for his helpful discussion and encouragement, Abbas Movahhedi for pointing out an error on p-rationality in the first version of the paper, and Christian Maire for Remark 6.23. We also would like to thank Jilali Assim, Zouhair Boughadi, Asuka Kumon, El Boukhari Saad, Bouchaib Sodaigui, and Youness Mazigh for their interest in this work and helpful comments. The revision of this paper was carried out while D. Lim was a visiting researcher at FEMTO-ST in Besancon in 2023 and during his visit to the Institute for Advanced Studies in Mathematics (IASM) in Harbin Institute of Technology in 2025. D. Lim would like to thank both institutions for their friendly environment. Finally, we would like to thank the anonymous referee for his or her patience over a long period and for the sharp and constructive comments, which greatly enhanced the overall quality of the paper. In particular, we are grateful for the valuable mathematical suggestions regarding Section 4.