= Prime ideals and local embedding orbits
For a finite extension of <number fields> $L/K$ and a nonzero <prime ideal> $\mathfrak p$ of $K$, the primes of $L$ above $\mathfrak p$ correspond to the orbits of
$$
\operatorname{Gal}(\overline K_{\mathfrak p}/K_{\mathfrak p})\quad\text{on}\quad\operatorname{Hom}_K(L,\overline K_{\mathfrak p}).
$$
An embedding determines a prime by pulling back the local <absolute value on a field>. Embeddings with the same prime extend to embeddings of the same completion and are conjugate. If $L=K(\alpha)$, these orbits are the irreducible factors of its <minimal polynomial> over $K_{\mathfrak p}$.
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