Trees and Solvability of Depressed Cubics over $$\mathbb{Q}_{2}$$

We describe properties of 2-adic valuation trees associated to depressed cubic polynomials. By doing so, we provide a criterion for the solvability of a depressed cubic equation and the number of solutions over some domains of $$\mathbb{Q}_2$$ . This leads to the locality of the solutions of the depressed cubic equation over $$\mathbb{Q}_2$$ . We also present an example illustrating how the dynamics of depressed cubic polynomials with infinite tree branches may fail to be topologically transitive (i.e., dense) over the set $$\mathbb{Z}_2$$ .