Growing of the Artin conductor

By using geometry of numbers, Minkowski showed that there exists a constant $C$ such that if $D_K$ is the discriminant of a number field $K$, then $\vert D_K \vert >C^{\left[K:Q\right]}$. In 1978, from the existence of infinite class field towers, Martinet constructed sequences of number fi elds of growing degree and bounded root discriminant. It is natural to ask if it is possible to extends the previous results to the Artin conductor. In 1977 Odlyzko, found the first nontrivial lower bounds for the conductor and in 2011 by using analytic methods, we improved these bounds. In this talk, we will show the existence of irreducible Artin characters of growing degree with bounded root conductors.