Integration on Valuation Fields over Local Fields

Constructing class fields over local fields

Let K be a p-adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L/K of degree p to the exponents of generators of the norm group NL/K(L ∗). This is applied in an algorithm for the construction of class fields of degree pm, which yields an algorithm for the computation of class field...

Semigroups over Local fields

Let G be a 1-connected, almost-simple Lie group over a local field and S a subsemigroup of G with non-empty interior. The action of the regular hyperbolic elements in the interior of S on the flag manifold G/P and on the associated Euclidean building allows us to prove that the invariant control set exists and is unique. We also provide a characterization of the set of transitivity of the contr...

Bernstein's Theorem over Fields with Discrete Valuation

Computing Tropical Varieties over Fields with Valuation

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

Quadratic Forms over Discrete Valuation Fields

As a general rule, the theory of quadratic forms over a ring R goes much more smoothly if R is non-dyadic. Of course, if R is a field then this simply says that we wish to avoid characteristic 2, but in general there is more to it than this. For instance, the ring Z is dyadic according our definition whereas for an odd prime power q Fq[t] is not, and indeed the theory of quadratic forms is some...