Some quadratically closed fields of nimbers

Lucia Risnoveanu

mathematical logic

Some quadratically closed fields of nimbers

arXiv:2606.28414

summary

The paper studies Nim arithmetic on ordinals and identifies the next quadratically closed field ordinals after ε₀, showing that the set {ε_α | α ≤ Ω^{Ω^Ω}} forms this next class.

Abstract

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 $I ¨ ^{I ¨ ^{I ¨} }$ . The problem of finding the next transcendental ordinal is still open. Two years later, H. Lenstra proved that $ε_{0}$ is the next quadratically closed field ordinal. In this paper, we prove that ${ε_{I^} \pm ∣ I^\pm \leq I ¨ ^{I ¨ ^{I ¨} }}$ are the next quadratically closed field ordinals.

Topics & keywords

#nimbers #ordinal arithmetic #quadratically closed fields #epsilon numbers #transcendental ordinalsNim arithmeticquadratically closed fieldε₀ε_αΩ^{Ω^Ω}ConwayLenstra