Solving quadratic equations over polynomial rings of characteristic two

We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain $A$ with 1 and a polynomial equation $a_n\,t^n+\cdots +a_0=0$ with coefficients $a_i$ in $A$, our problem is to find its roots in $A$.

We show that when $A=B[x]$ is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over $B$. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over $A$ when $A$ is $F[x_1,\ldots,x_N]$ or $F(x_1,\ldots,x_N)$ for any finite field $F$ and any number $N$ of variables.

The case of quadratic equations in characteristic two is studied in detail.