# Thesis Defense: Jesse Patsolic

Posted on: April 21, 2014

**Trinomials Defining Quintic Number Fields**

**Jesse Leigh Patsolic**

**Friday, April 25th at 3:30 pm. Manchester 20.**

Given a number field K, how does one find polynomials f(x), with a root in K, that have a small number of non-zero terms? Is it possible to make this method work to classify all the trinomials that generate a given field?

We start by computing the genus four curve, C_{K}, that parameterizes the trinomials defining K. a specific degree five number field K. We then compute a map from C_{K} to a cubic curve E. In the case where K is generated by f(x) = x^{5} + x + 3, the curve E is an elliptic curve, with positive rank, defined over a degree ten number field L. We discuss the method used to compute the map from C_{K} to E. For future work, we hope to find generators of the set E(L), to provably conclude, up to equivalence, the trinomials that define K, as mentioned above.