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We give two specializations of Krasner's mass formula. The first formulayields the number of extensions of a $\mathfrak{p}$-adic field with given,inertia degree, ramification index, discriminant, and ramification polygon. Wethen refine this formula further to the case where an additional invariantrelated to the residual polynomials of the segments of the ramification polygonis given.

Counting Extensions of $\mathfrak{p}$-Adic Fields with Given Invariants