In On Numbers And Games, Conway uses the term "genetic" for definitions of operations on surreal numbers that are inductive in terms of their options. His definitions of addition and multiplication are genetic, and he also gives genetic definitions of division and square roots, though the latter are more complicated because the options of each result are inductively defined separately from the overall induction over surreals.
Conway also shows that the surreal numbers are a real-closed field, i.e. that every polynomial of odd degree with surreal coefficients has a surreal root. However, this proof is not genetic, but relies on convergence of infinite series.
My question is: is there a genetic proof of the real-closure of the surreal numbers? That is, given a surreal polynomial of odd degree, can I define a root of that polynomial in terms of its options, by surreal induction on the options of the coefficients (and presumably also some kind of "inner" induction in the options, as in the case of division and square roots)? Note that division and square roots are particular instances of roots of polynomials, namely $a x = 1$ and $x^2 = a$.
(One might initially guess that this is unlikely because a given polynomial of odd degree will generally have many roots with no obvious way to canonically choose one, so there couldn't be a completely canonical construction of a root. However, the construction doesn't have to be completely canonical; it could involve an arbitrary "initial value" choice, such as the first input to Newton's method, after which the "inner" inductive definition of the options successively "converges" to some root determined by the initial value. So it doesn't seem impossible to me that such a genetic construction could exist.)
