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The Newton polygon of an Eisenstein polynomial is a straight line of slope $1/n$, as shown in the picture above. A single kitten—Nyatta, the character from Nekojiru’s manga—perches on that line.

For a polynomial $f(x)$, the condition that its Newton polygon is a single straight segment is equivalent to all of its roots having $p$-adic valuations equal to the slope of that segment (see Neukirch, ''Algebraic Number Theory'', Proposition 6.3).

Leveraging this equivalence, one can prove the irreducibility of an Eisenstein polynomial over a $p$-adic field purely via its Newton polygon—without appealing to Gauss’s lemma, which says that a polynomial irreducible over an integrally closed domain remains irreducible over its field of fractions.