Different exponent and tame ramification ($d_{\mathfrak{P}}\ge e_{\mathfrak{P}}-1$)

For a finite extension of number fields, the different exponent satisfies $d_{\mathfrak{P}}\ge e_{\mathfrak{P}}-1$, with equality exactly when the extension at $\mathfrak{P}$ is tamely ramified. The finite residue fields are perfect, so tameness is equivalent to the residue characteristic not dividing the ramification index. Thus unramified primes have exponent zero and wildly ramified primes have exponent at least $e_{\mathfrak{P}}$. No Galois extension hypothesis is needed.