= Restriction and norm compatibility of the Artin map
{title2=$\operatorname{Art}_{LE/E}(\mathfrak a)|_L=\operatorname{Art}_{L/K}(N_{E/K}\mathfrak a)$}
For an <abelian extension> $L/K$ of <number fields>, restriction to an intermediate <field> takes each <Artin symbol> to that of the smaller extension. For a finite base extension $E/K$, restriction identifies $\operatorname{Gal}(LE/E)$ with $\operatorname{Gal}(L/(L\cap E))$. At an unramified prime $\mathfrak q$ of $E$ above $\mathfrak p$, its residue-field Frobenius restricts to the $f(\mathfrak q/\mathfrak p)$th power of the <Frobenius automorphism> at $\mathfrak p$. Since $N_{E/K}\mathfrak q=\mathfrak p^{f(\mathfrak q/\mathfrak p)}$, the <group homomorphism> property proves the displayed identity for <fractional ideals> supported over unramified primes.
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