Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ordered field No of surreal numbers. Is the field No algebraic over the field $K(\Omega)$ (and therefore equal to the real closure of $K(\Omega)$)? If not, what is the transcendence degree of No over $K(\Omega)$?

EDIT: I think on second thought that the real closure of $K(\Omega)$ intersected with $\mathbb{R}$ in No is possibly just the real closure of $\mathbb{Q}$, rather than all of $\mathbb{R}$. Perhaps then I need to consider the real closure of the compositum in No of $K(\Omega)$ and $\mathbb{R}$ and ask if No is algebraic over that. If that's not true, then is there some way of bootstrapping such a construction to some sort of algebraic construction of No from $K(\Omega)$?

Related mathoverflow post:

Will Sawin's answer to another mathoverflow question shows that No is a proper extension of $K(\Omega)$. See Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

It is perhaps relevant to note that $K[\Omega]$ can be identified with the polynomial ring $\mathbb{Z}[\omega^{\omega^\alpha}: \alpha \in \Omega]$ generated by the (algebraically independent) delta numbers $\omega^{\omega^\alpha}$ for $\alpha \in \Omega$, and the field $K(\Omega)$ is then identified with the field $\mathbb{Q}(\omega^{\omega^\alpha}: \alpha \in \Omega)$ of rational functions in the delta numbers.

ADDITION TO ORIGINAL QUESTION:

Since in NBG the class $\Omega$, and therefore also the class $\{\omega^{\omega^\alpha}: \alpha \in \Omega\}$, maps onto every class, it follows that the ring $K[\Omega]$ maps homomorphically onto every commutative ring, even those with an underlying proper class. In particular, there is a surjective ring homomorphism $K[\Omega] \longrightarrow \operatorname{No}$, or equivalently a maximal ideal $M$ in $K[\Omega]$ such that $K[\Omega]/M$ is isomorphic to $\operatorname{No}$. Is there a nice way to define such a homomorphism and/or maximal ideal?

YET ANOTHER ADDITION: Is $K[\Omega]$ isomorphic to a quotient ring of the ring Oz of omnific integers?