On K-derived quartics and invariants of local fields

This dissertation will cover two topics. For the first, let $K$ be a number field. A $K$-derived polynomial $f(x) \in K[x]$ is a polynomial that

factors into linear factors over $K$, as do all of its derivatives. Such a polynomial

is said to be {\it proper} if

its roots are distinct. An unresolved question in the literature is

whether or not there exists a proper $\Q$-derived polynomial of degree 4. Some examples

are known of proper $K$-derived quartics for a quadratic number field $K$, although other

than $\Q(\sqrt{3})$, these fields have quite large discriminant. (The second known field

is $\Q(\sqrt{3441})$.) I will describe a search for quadratic fields $K$

over which there exist proper $K$-derived quartics. The search finds examples for

$K=\Q(\sqrt{D})$ with $D=...,-95,-41,-19,21,31,89,...$.\\

For the second topic, by Krasner's lemma there exist a finite number of degree $n$ extensions of $\Q_p$. Jones and Roberts have developed a database recording invariants of $p$-adic extensions for low degree $n$. I will contribute data to this database by computing the Galois slope content, inertia subgroup, and Galois mean slope for a variety of wildly ramified extensions of composite degree using the idea of \emph{global splitting models}.