= Different exponent and tame ramification
{title2=$d_{\mathfrak P}\geq e_{\mathfrak P}-1$}
For a finite extension of <number fields>, the <different exponent> satisfies $d_{\mathfrak P}\geq e_{\mathfrak P}-1$, 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.
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