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Norm Form Equations

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

Abstract

In this chapter we shall require some knowledge about algebraic number fields. Throughout, K will be an algebraic number field of degree k, and ℜ will denote the norm from K to the field ℚ of rationals. There are k isomorphic embeddings ϕ1,...,ϕk, of K into the complex numbers; denote the image of an element α of K under ϕi by ϕ(i). We will always tacitly assume that ϕ1, is the identity map, so that α (1) = α, and we shall say that α (1) = α, α (2),...α (k) are the conjugates of α. In this notation ℜ(α) = α (1)...α (k). Given a linear form
$$ M(\mathop X\limits_ = ) = M(X_1 , \ldots ,X_n ) = \alpha _1 X_1 + \cdots + \alpha _n X_n , $$
(1.1)
with coefficients in K we write
$$ \Re (M(\mathop X\limits_ = )) = \prod\limits_{i = 1}^k {M^{(i)} (X_1 , \ldots ,X_n )} = \prod\limits_{i = 1}^k {(\alpha _1^{(i)} X_1 + \cdots + \alpha _n^{(i)} X_n )} . $$

Keywords

Linear Form Finite Index Real Form Primitive Element Integer Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Z.I. Borevich and I.R. Shafarevich (1966). Number Theory. (Translated from the Russian). Academic Press.Google Scholar
  2. W.M. Schmidt (1971c). Linearformen mit algebraischen Koeffizienten. II. Math. Ann. 191, 1–20.zbMATHCrossRefMathSciNetGoogle Scholar
  3. ____ (1972). Norm form equations. Ann. of Math. 96, 526–551.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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