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Auxiliary Polynomials in Number Theory by David Masser

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14

Heights

As the exercises in the previous chapter show, we are now in a good position to prove the Gelfond–Schneider Theorem on αβ (and much more). But the reader (not to mention the writer) deserves a rest from all this intricate estimating. So before going further with transcendence we will introduce another way of measuring algebraic numbers α. The norm ||α|| defined in (11.3) of Chapter 11 is quite useful, but essentially only for algebraic integers, and we would like to take denominators into account, while at the same time preserving handy properties like (11.6). We will define a height H(α) which also has the important property, sometimes associated with the name of Northcott, that for any d and any H there are at most finitely many ...

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