Defining polynomial of a local field (reviewed)

The defining polynomial of a $p$-adic field $K$ is an irreducible polynomial $f(x)\in\mathbb{Q}_p[x]$ such that $K\cong \mathbb{Q}_p(a)$, where $a$ is a root of $f(x)$.

The defining polynomial can be chosen to be monic with coefficients in $\mathbb{Z}_p$; by Krasner's lemma, we can further take $f(x)\in \mathbb{Z}[x]$.

The LMFDB uses the following conventions for choosing defining polynomials:

1. For unramified extensions, the polynomial needs to be irreducible modulo $p$. When it is feasible to compute, we use the Conway polynomial for the residue prime $p$ and the given degree.
2. For totally ramified extensions, we pick an Eisenstein polynomial which is reduced in the sense of Monge.
3. In the remaining cases, we pick a polynomial which is in Eisenstein form.