The fourth paragraph of section 5 of F Richman's paper Var Der Waerden's construction of a splitting field claims that the universal splitting algebra of a monic polynomial over a field need not itself be a field.
I am confused by this. Firstly, the extension $A[s_1,\dots ,s_n]\subset A[x_1,\dots ,x_n]$ is integral (the $s_i$ are the elementary symmetric polynomials). Given a monic $f\in A[x]$, evaluating the above extension at the coefficients of $f$ gives the universal splitting algebra $A\to \mathrm{Split}_A(f)$, which is therefore also integral. But then, if $A$ is a field so must be the splitting algebra. What am I missing?
An integral extension of a field which is a domain must be a field. But the universal splitting algebra may not even be a domain. For instance, if $f$ is a quadratic then the universal splitting algebra will just be $A[x]/(f)$ (since once you have one root of $f$ the other is uniquely determined), which will only be a domain if $f$ is irreducible.