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Let $L/K$ be a finite abelian extension and let $\mathfrak{m}$ be an admissible modulus, so every ramified prime divides ${\mathfrak{m}}_0$ and the Artin reciprocity mapis defined. If $\mathfrak{p}\nmid {\mathfrak{m}}_0$, then the ideal-theoretic decomposition law says that its residue degree isand, since $\mathfrak{p}$ is unramified, it hasprime factors in $L$.

Indeed, ${\theta }_{L/K}(\mathfrak{p})$ is the Artin symbol, whose restriction to the residue field is $x\mapsto x^{N\mathfrak{p}}$. The Galois group of the finite residue-field extension is generated by this Frobenius automorphism and has order $f(\mathfrak{P}/\mathfrak{p})$. The fundamental identity $efg=[L:K]$, with $e=1$, gives the formula for $g$.

For an unramified prime $\mathfrak{p}$, the ideal-theoretic decomposition law givesUnder the Hilbert class field isomorphism, this Artin symbol is the image of $[\mathfrak{p}]\in \mathrm{Cl}(K)$. It is trivial exactly when $[\mathfrak{p}]=1$, which means exactly that $\mathfrak{p}$ is a principal ideal.