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Simultaneous Approximation to Algebraic Numbers

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Part of the Lecture Notes in Mathematics book series (LNM,volume 785)

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,...,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.

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References

  • W.M. Schmidt (1970). Simultaneous approximation to algebraic numbers by rationals. Acta Math. 125, 189–201.

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  • ____ (1971b). Linear forms with algebraic coefficients. I. J. of Number Th. 3, 253–277.

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  • ____ (1971c). Linearformen mit algebraischen Koeffizienten. II. Math. Ann. 191, 1–20.

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© 1980 Springer-Verlag Berlin Heidelberg

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(1980). Simultaneous Approximation to Algebraic Numbers. In: Diophantine Approximation. Lecture Notes in Mathematics, vol 785. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38645-2_6

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  • DOI: https://doi.org/10.1007/978-3-540-38645-2_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09762-4

  • Online ISBN: 978-3-540-38645-2

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