Let $R$ be a subring of $\mathbf{No}$, the set of surreal number. We try to construct $\tilde{R}$, the Cauchy completion of $R$, just like the ordinary Cauchy completion for metric space.

In the following we only consider the sequences in $R$ indexed by (i.e. with length equal) $\mathrm{Cf}(R)$, the cofinality of $R$. For any Cauchy sequences $(x_{\alpha})$ and $(y_{\alpha})$, we define an equivalent relation: $$x\sim y\;\;\mathrm{ iff }\;\; |x_{\alpha}-y_{\alpha}|\rightarrow0.$$ Let $\tilde{R}$ be the set of all equivalent classes of Cauchy sequences. On $\tilde{R}$ we define addition $[x]+[y]=[x+y]$ and multiplication $[x][y]=[xy]$ of classes. It is standard to check that these operations are well defined, $\tilde{R}$ becomes a ring and is Cauchy complete, and that $R$ is dense in $\tilde R$.

For each ordinal number $\alpha$, denote $O_{\alpha}$ the set of surreal numbers with birthday $<\alpha$. It is known that if $\alpha=\omega^{\omega^{\beta}}$ for some ordinal $\beta$ then $O_{\alpha}$ is a ring, and if $\alpha=\epsilon_{\beta}$ for some ordinal $\beta$ then $O_{\alpha}$ is a field. It is easy to check that in the latter case, $\tilde{O_{\alpha}}$ is not only ring but also a field. The question is, in the case $\alpha=\omega^{\omega^{\beta}}$, is $\tilde{O_{\alpha}}$ actually a field?

It is worth pointing out that if $\beta=0$ then $O_{\alpha}$ is the set of dyadic fraction and hence $\tilde{O_{\alpha}}=\mathbf R$, the set of reals, and is certainly a field. Apparently the difficult part is about the existence of multiplicative inverse.