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- W4299578001 abstract "AbstractThe 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,...,qd−1, p are rational integers and where q = max(|q1|,...,|qd−1|) ≠ 0. Taking qv+1 = ... = qd−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,...,qv, p having q = max(|q1|,..., |qv|) > 0." @default.
- W4299578001 created "2022-10-02" @default.
- W4299578001 date "2009-02-04" @default.
- W4299578001 modified "2023-10-05" @default.
- W4299578001 title "Simultaneous Approximation to Algebraic Numbers" @default.
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- W4299578001 doi "https://doi.org/10.1007/978-3-540-38645-2_6" @default.
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