Integral trace need not imply integrality
= Integral trace need not imply integrality
{title2=$\operatorname{Tr}(x)\in\mathcal O_K\ \not\Longrightarrow\ x\in\mathcal O_L$}
In $L=\mathbb Q_p(\theta)$ with $\theta^2=p$, the <Eisenstein polynomial> $X^2-p$ gives $v_L(\theta)=1$ and $v_L(p)=2$. The element $x=\theta/p$ has <field trace> zero, but <valuation> $-1$, so is not in the <valuation ring>. Cancellation in a <field trace> contrasts with the multiplicative test in <unit norm detects units in a local extension>.