= Ramification break of an Artin–Schreier pole
{title2=$b_{\rm lower}=b_{\rm upper}=m$}
Let $K$ be an equal-characteristic-p <local field> and let $c\in K$ have $v_K(c)=-m<0$, with $p\nmid m$. If $\alpha^p-\alpha=c$, the <Artin–Schreier extension> $L=K(\alpha)$ has degree $p$ and is totally ramified: a root in $K$ would force the pole order to be divisible by $p$, and $p v_L(\alpha)=-m e(L/K)$ forces $e=p$. Choose $1\leq s<p$ and an integer $r$ with $rp-sm=1$. For a base <uniformizer> $t$, the element $\pi_L=t^r\alpha^s$ has <valuation> one. Under $\sigma_a(\alpha)=\alpha+a$, $a\in\mathbb F_p^\times$, the leading term in $\sigma_a(\pi_L)-\pi_L$ has <valuation> $rp-(s-1)m=m+1$. The <uniformizer criterion for lower ramification groups> therefore gives the single lower break $m$. The <Herbrand function> is the identity up to that break, so the upper break is also $m$.
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