Artin symbol ($\left(\frac{L/K}{\mathfrak{p}}\right)$)

For an unramified prime $\mathfrak{p}$ in an abelian extension $L/K$, the Artin symbol is the unique Galois element acting on residue fields by $x\mapsto x^{N\mathfrak{p}}$.

The Artin map sends an ideal prime to the ramified primes to the product of its Frobenius elements. For a suitable modulus, its kernel is the norm or congruence subgroup associated with the abelian extension.