Simultaneous Approximation to Algebraic Numbers

Abstract

The only easy result here is as follows. Suppose^†) 1,α ₁,...,α _v are algebraic and linearly independent over the rationals. Let d be the degree of the number field generated by α ₁,...,α _v and let 1,α ₁,...,α _v,...,α _d−1 be a basis of this field. We saw in Theorem 4A of Chapter II that α ₁,...,α _d−1 are badly approximable, so that

$$ \left| {\alpha _1 q_1 + \cdots + \alpha _{d - 1} q_{d - 1} - p} \right| > c_1 q^{ - d + 1} $$

where q₁,...,q_d−1, p are rational integers and where q = max(|q₁|,...,|q_d−1|) ≠ 0. Taking q_v+1 = ... = q_d−1 = 0, we have LEMMA 1A. Suppose 1,α ₁,...,α _v are linearly independent over ℚ, and they generate an algebraic number field of degree d. Then

$$ \left| {\alpha _1 q_1 + \cdots + \alpha _v q_v - p} \right| > c_1 q^{ - d + 1} $$

for arbitrary integers q₁,...,q_v, p having q = max(|q₁|,..., |q_v|) > 0.