## On K-derived quartics and invariants of local fields

Abstract 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 thatfactors into linear factors over $K$, as do all of its derivatives. Such a polynomial is said to be {\it proper} ifits 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 find... (more) 2019 Carrillo, Benjamin (Author) / Jones, John (Advisor) / Bremner, Andrew (Advisor) / Childress, Nancy (Committee member) / Fishel, Susanna (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher) Mathematics Doctoral Dissertation 2610 pages English Doctoral Dissertation Mathematics 2019 Graduate College / ASU Library MODS / OAI Dublin Core / RIS

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