Some quadratically closed fields of nimbers

In 1976, J. H. Conway introduced Nim arithmetic which establishes an algebraically closed field structure over the class of ordinals and proved that the first transcendental ordinal is $\omega^{\omega^\omega}$. The problem of finding the next transcendental ordinal is still open. Two years later, H. Lenstra proved that $\varepsilon_0$ is the next quadratically closed field ordinal. In this paper, we prove that $\{\varepsilon_\alpha \mid \alpha \leq \omega^{\omega^\omega} \}$ are the next quadratically closed field ordinals.