Classifying extensions of the field of formal Laurent series over $\mathbb{F}_p$

In previous works, Jones and Roberts and Pauli and Roblot have studied finite extensions of the $p$-adic numbers $\mathbb{Q}_p$. This paper focuses on results for local fields of characteristic~$p$. In particular, we are able to produce analogous results to Jones and Roberts in the case that the characteristic does not divide the degree of the field extension. Also, in this case, following from the work of Pauli and Roblot, we prove that the defining polynomials of these extensions can be written in a simple form amenable to computation. Finally, if $p$ is the degree of the extension, we show there are infinitely many extensions of this degree and thus these cannot be classified in the same manner.