= Branch-cycle conjugacy class
{title2=$[\tau_p],\qquad\tau_p(\pi)/\pi\equiv\zeta_{e_p}$}
For a <Finite Galois extension> of $K(T)$ with algebraically closed characteristic-zero constants, the local decomposition and inertia groups coincide and are cyclic. A fixed compatible choice of primitive <roots of unity> selects a generator through its <uniformizer character of inertia>. Changing the prime above a base point conjugates this generator, and changing the <uniformizer> does not change its character. Without the roots-of-unity convention, only the inertia-subgroup <conjugacy class> is intrinsic. Over $\mathbb C$, positive local orientation gives the standard convention.
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