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I think p-adic fields themselves are somewhat chimeric. Although I know better, I can never fully avert the tendency to think of them as having characteristic p, rather than zero. Indeed, I just heard a number theorist refer to them as being of "mixed characteristic", meaning that although $\mathbb{Z}_p$ has characteristic zero, its residue field is $\mathbb{F}_p$ has characteristic p. I understand that this allows you to pass information from the Galois groups of finite fields (whose elements can be explicitly identified using Frobenius maps that only make sense in positive characteristic) to Galois groups of local fields, and thence to Galois groups of global fields.