ON INTEGRAL MODELS OF ALGEBRAIC TORI AND AFFINE TORIC VARIETIES B. ` E. KUNYAVSKI ˘ I AND B.Z. MOROZ To the memory of Albrecht Fr¨ohlich §1. Introduction In a joint work with V.E. Voskresenski˘ ı [14], an explicit construction of a natural integral model of an algebraic torus, defined over a number field, has been described; in [13], we have constructed a few integral models of the affine toric varieties associated to such a torus. Inspired by a paper of A. Fr¨ ohlich’s [7], we have conjectured [14] that for any number field k there is a sufficiently big finite normal extension L | k having an integral basis over k, and pointed out that our constructions of integral models could be considerably simplified under this conjecture [14]. In the meantime M. V. Bondarko [2] has proved the conjecture. One of the goals of this paper is to describe the arising simplifications in some detail. Our second goal is to restore a few details of our argument and notation, left to the reader in [13] and [14]. We do not dwell on the applications of integral models here; some of the applications have been discussed in [13], [14] (cf. also [5]). In the next section we shall collect a few results and definitions, relating to the theory of algebraic tori and affine toric varieties defined over an arbitrary field of characteristic 0. Although both our definition of an affine T - toric variety and our Theorem 1 may be known to some authors, we could not find a proof of that theorem in the literature. After the proper terminology has been established, it is a relatively straightforward matter to construct our “standard” model of an algebraic torus T defined over a number field, and the corresponding models of the T -toric varieties. This is done in Section 3. Notation and conventions. As usual, Q, Z, N, and F p stand for the field of rational numbers, the ring of rational integers, the monoid of non- negative rational integers, and the finite field of p elements respectively. The algebraic closure of a field k and the degree of a finite extension of fields L | k are denoted, respectively, by ¯ k and [L : k]. Given a commutative ring A, let A * stand for the group of invertible elements in A, and let G m,A 1