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Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p < n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer's one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in [1 ]. [ABSTRACT FROM AUTHOR]

Let F(x,y) = ax3 + bx2y + cxy2 + dy3 ∊ Z[x,y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that V2(k) ≡ 0 (mod 3) and 2v2(--2b3 -- 27a2d + 9abc) = 3v2(b2 -- 3ac), we prove that the cubic Thue equation F(x, y) = k has no solution (x, y) E Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x, y). [ABSTRACT FROM AUTHOR]