| Summary: | Local Fields, complete discrete valuation fields with a finite residue field, are an important mathematical tool in number theory; in particular in the topics of Class Field Theory and Arithmetic Geometry. However, while a lot of work has been done on when the residue field, $\overline{F}$, of the complete discrete valuation field, $F$, is finite; dealing with less restrictions on the fields is also an incredibly fruitful undertaking. This enterprise, both examining some of the work already done on this topic and also expanding on them, is the focus of the thesis.
In this thesis we look at two important aspects of the theory of complete discrete valuation fields. To begin with we examine local class field theory on complete discrete valuation fields whose residue field is an imperfect field of positive characteristic $p$. We investigate for which finite abelian totally ramified $p$-extensions $L/F$ is the map, $\Psi_{L/F}$, an isomorphism; as before we only knew it was an isomorphism when $L/F$ was a finite cyclic extension.
We then move onto abelian varieties, $A$, over complete discrete valuation fields. The research on this topic focuses on generalising a result by Barry Mazur about the rational points of an abelian variety with good ordinary reduction over a complete discrete valuation field with finite residue field, $A(F)$, to also deal with complete discrete valuation fields whose residue field is perfect and has positive characteristic $p$.
We finish off the thesis by then bringing up further directions to take both topics in the future. The work here opens up several new avenues in the subjects to explore and plenty of new research opportunities to be investigated later on.
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