Univerzální kvadratické formy nad řády v číselných tělesech

Krásenský, Jakub

This thesis studies quadratic forms and lattices over rings of integers in number fields, and, to some extent, over non-maximal orders as well. The main focus is on universality of forms and lattices, and on the connected notion of the Pythagoras number. We mostly study totally positive definite forms and lattices over totally real fields, as this is arguably the most difficult case with very unpredictable behaviour. In some chapters we develop a general theory valid for fields of all degrees - in particular when we study the quadratic Waring's problem -, while in others we present important detailed results for families of fields in low degrees. By this we fulfill several purposes: We obtain interesting results, directly in line with questions studied by the likes of Siegel; we illustrate and further develop a multitude of techniques which could be applied in other situations; and we formulate conjectures and questions which can stimulate further research. Particular focus is put on real biquadratic fields; the first two chapters significantly advance the theory of integral quadratic forms over them. 1

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