An Artin--Mumford criterion for conic bundles in characteristic two

We prove a characteristic two version of the famous criterion of Artin and Mumford for irrationality of conic bundles. On the one hand, combined with the pathological behaviour of conic bundles in characteristic two, this allows us to construct easier and more explicit examples of irrational conic bundles. On the other hand, degeneration techniques \`a la Voisin allow to deduce irrationality results in characteristic zero.