Splitting probability invariant for Cordes schemes

Abstract

Let $S=(G,-1, D_G)$ be a Cordes scheme. In this paper, along with generalizing the results of splitting probability invariant of fields to Cordes schemes, it is also shown that $p_s(S)>1/2$ if and only if S is a generalized Hilbert scheme except when S is a scheme of a formally real field with $|G/R(S) |=4$, where R(S) denotes the radical of S. We also prove that if $S_1, S_2$ are generalized Hilbert schemes, $P(S_i)=\{p_s(S_i^{T_n}) : |T_n |= 2^n \text { and } n \in \mathbb {N}\}$; $i=1, 2$ and $|P(S_1) \cap P(S_2) |> 1$, then $P(S_1)=P(S_2)$.