conductor of an abelian extension

《Local Fields》 · 第 228 页 · Chapter XV，Section 2「The Conductor of an Abelian Extension」

Let $c$ be the largest integer for which $G^c\ne\{1\}$, and put $f=\varphi(c)+1$. Then $U_K^f\subset N_{L/K}(L^*)$, and $f$ is the least integer with this property. The ideal $\mathfrak p_K^f$ is called the conductor of $L/K$.