Exploring the Connections between Polynomials, Field Theory, and Roots of Unity

In this talk, we will delve into the historical and mathematical connections between polynomial equations, field theory, and roots of unity. Through simple yet insightful examples, we will highlight the significance of recent research advancements. Specifically, we will discuss the concepts of radical closure, Artin-Schreier closure, and elementary abelian closure, which we introduced in collaboration with Kenneth Ward for the two first concepts, Mpendulo Cele for the last concept and further investigated alongside Elizabeth Mrema, Chad Brache Valentijn Karemaker, and Jeroen Sijsling. The talk will shed light on the profound role of roots of unity within field theory, supported by our collaborative work with Elizabeth Mrema. As we unravel the captivating interplay among polynomials, field theory, and roots of unity, we will unveil exciting research directions and their significance in contemporary mathematics.

References

1. Generic polynomials for cyclic function field extensions over certain finite fields. European Journal of Mathematics, 4(2):585-602, link (2017).
2. Cubic fields: a primer (with Kenneth Ward), European Journal of Mathematics, link (2018).
3. An explicit triangular integral basis for any separable cubic extension of a function field (with Kenneth Ward), European Journal of Mathematics, link, (2018).
4. A complete study of the ramification for any separable cubic global function field, (with Kenneth Ward) Research in Number Theory 5, 36, link (2019).
5. The Geometry Of The Moduli Space Of non-cyclic Biquadratic Field Extensions. (with Mpendulo Cele). Quaestiones Mathematica, link (2021)
6. Cubic function fields with prescribed ramification. (with Valentijn Karemaker and Jeroen Sijsling). International Journal of Number Theory. Vol. 17, No. 09, pp. 2019-2053, link (2021)
7. Quadratic cyclotomic moduli spaces. (with Elizabeth Mrema). link 2022

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