Abstract
Let $K$ be a local field and $k$ an algebraically closed field. We prove the finiteness of isomorphism classes of semisimple Galois representations of $K$ into $\GL_d(k)$ with bounded Artin conductor and residue degree. We calculate explicitly the number of totally ramified finite abelian extensions of $K$ with bounded conductor. Using this result, we give an upper bound for the number of certain Galois extensions of $K$.
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