= Ramification breaks from a reduced ramification polygon
{title2=$\operatorname{length}(-m)=|G_m|-|G_{m+1}|$}
In a totally ramified <Galois extension>, the <uniformizer criterion for lower ramification groups> gives $\sigma\in G_m\setminus G_{m+1}$ exactly when $v_L((\sigma(\pi)-\pi)/\pi)=m$. The <Newton polygon root valuation theorem> therefore identifies a slope $-m$ in the coefficient-index convention with a lower ramification jump at $m$. Its horizontal length is the number of automorphisms in that difference. Under the <reflected Newton polygon convention> the same jump has slope $+m$.
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