On ther-rank Artin Conjecture, II

On primitive points of elliptic curves with complex multiplication

“…Then L ℘ 0 is a quadratic extension over k. Let 1 represent the density of splitting primes which do not split completely in L ℘ 0 , or any field k ℘ , ℘ ∈ I . Then one has 1 . So it is enough to show 1 > 0.…”

Divisibility of reduction in groups of rational numbers

“…(2) The conditions that either m is odd or that gcd(m, σ Γ S ,m ) = 1 in the statement of Theorem 3 can be removed at the cost of complicating the expression for χ Γ S ,m . (3) It was proven in [1] that if Γ ⊂ Q * is a finitely generated subgroup, the Generalized Riemann Hypothesis implies that the set of primes for which ind p (Γ) = 1 has a density δ Γ that equals…”

“…, a r ). We also set ∆ k = ∆ k (Γ) = 1 for k ≤ 0 and ∆ k = ∆ k (Γ) = 0 for k > r. It can be shown (see [1,Section 3]) that ∆ 1 , . .…”

On a problem of Schinzel and Wójcik involving equalities between multiplicative orders

“…we need to employ Kummer Theory (see Lang book [4, Chapter VIII, section 8] and also [1]) that allows us to deduce the next result: LEMMA 7. Let M 1 be an integer.…”

“…The invariant s ( ) of a multiplicative subgroup ⊆ Q * with rank Z ( ) = s, is defined as the greatest common divisor of all the minors of size s of the relation matrix of the group of absolute values of (see [1,Section 3·1] for some details).…”