Relative Spectral Norm on Algebraic Numbers

On solving relative norm equations in algebraic number fields

Let Q ⊆ E ⊆ F be algebraic number fields and M ⊂ F a free oE -module. We prove a theorem which enables us to determine whether a given relative norm equation of the form |NF/E (η)| = |θ| has any solutions η ∈ M at all and, if so, to compute a complete set of nonassociate solutions. Finally we formulate an algorithm using this theorem, consider its algebraic complexity and give some examples.

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

Notes on Algebraic Numbers

This is a summary of my 1994–1995 course on Algebraic Numbers. (Revised and improved on 1993– 1994!) The background assumed is standard elementary number theory—as found in my Level III course—and a little (Abelian) group theory. Corrections and suggestions for improvement are welcome, and will be credited in future editions! I first learned algebraic number theory from Stewart & Tall’s book ([...

Algebraic Numbers and Algebraic Integers

c © W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

On approximation of real numbers by real algebraic numbers