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, CK, that parameterizes the trinomials defining K. a specific degree five number field K. We then compute a map from CK to a cubic curve E. In the case where K is generated by f(x) = x5 + 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 CK 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.