Branch-cycle conjugacy class ($[{\tau }_p],{\tau }_p(\pi )/\pi \equiv {\zeta }_{e_p}$)

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.

In characteristic zero with algebraically closed constants, a local unit has a square root. Thus $K(T)(\sqrt{a})$ near a point is equivalent to adjoining the square root of the local parameter to the power $v_p(a)$. Even order is unramified; odd order gives ramification index two. For a polynomial $a$, its order at infinity is minus its degree, so infinity branches exactly for odd degree.