Norm Form Equations

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


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
$$ M(\mathop X\limits_ = ) = M(X_1 , \ldots ,X_n ) = \alpha _1 X_1 + \cdots + \alpha _n X_n , $$
with coefficients in K we write
$$ \Re (M(\mathop X\limits_ = )) = \prod\limits_{i = 1}^k {M^{(i)} (X_1 , \ldots ,X_n )} = \prod\limits_{i = 1}^k {(\alpha _1^{(i)} X_1 + \cdots + \alpha _n^{(i)} X_n )} . $$


Linear Form Finite Index Real Form Primitive Element Integer Point 
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© Springer-Verlag Berlin Heidelberg 1980

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