The main part of this thesis comprises a study of quasilinear p-forms (i.e., Fermat-type forms of degree p over fields of characteristic p > 0) in the spirit of the theory of non-singular quadratic forms over fields of characteristic different from two. Our approach to the subject centres around the study of a basic discrete invariant of quasilinear p-forms known as the standard splitting pattern, which plays here a similar role to that played by Knebusch's splitting pattern invariant for non-singular quadratic forms in characteristic different from two. This invariant not only captures a certain notion of algebraic complexity for quasilinear p-forms, but also encodes essential information concerning the geometry of their projective zero-loci, which we call quasilinear p+hypersurfaces. We explore here various aspects of this interaction between the alge+ braic and algebra-geometric parts of the theory. Our approach is successful, not only in extending well-known results from the theory of non-singular quadratic forms over fields of characteristic different from 2 into this setting, but in revealing interesting and important new phenomena which are completely absent from the former theory. A second part of this thesis is concerned with studying the behaviour of mod+p Milnor K-groups of fields of characteristic p > 0 under scalar extension to the fW1ction fields of nowhere-smooth hypersurfaces. A unifying theme for all this work is the study of a basic birational invariant of hypersurfaces of the latter type, which we call the norm field.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:597111 |
| Date | January 2013 |
| Creators | Scully, Stephen |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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