= Surjectivity of decomposition group reduction
{title2=$D_{\mathfrak q}\twoheadrightarrow\operatorname{Gal}(k_{\mathfrak q}/k_{\mathfrak p})$}
For a finite <Galois extension> of <number fields>, choose a nonzero <residue field> generator at $\mathfrak q$ and lift it to an integral element $a$ whose residues at the other primes above $\mathfrak p$ are zero, using the <Chinese remainder theorem>. The orbit polynomial $\prod_{\sigma\in G}(T-\sigma a)$ has coefficients in the base <ring of integers>. Modulo $\mathfrak q$, its nonzero roots are precisely the orbit of $\overline a$ under the <decomposition group>. Its coefficients belong to $k_{\mathfrak p}$, so <finite-field Frobenius automorphism> permutes these roots. Consequently some decomposition-group element acts as Frobenius on the generator, and therefore on the entire <residue field>. Frobenius generates the finite-field <Galois group>, proving surjectivity.
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