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Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ isreduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists aparameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regularand $\nu(\mathfrak{m}/Q)\leq 1$. This leads to an affirmative answer to aproblem raised by Goto-Hong-Mandal. We also give an alternative proof (in facta strengthening) of their main result. In particular, we show that if$\widehat{R}$ is equidimensional, then $\bar{e}_1(Q)\geq 0$ for all parameterideals $Q\subseteq R$, and in characteristic $p>0$, we actually have$e_1^*(Q)\geq 0$. Our proofs rely on the existence of big Cohen-Macaulayalgebras.

Vanishing and non-negativity of the first normal Hilbert coefficient