In general, the value groups and the residue fields do not suffice to classify the algebraic henselian extensions of a valued field K, up to isomorphism over K. We define a stronger, yet natural structure which carries information about additive and multiplicative congruences in the valued field, extending the information carried by value groups and residue fields. We discuss the cases where these ``mixed structures'' give a solution of the classification problem.

Last update: February 3, 1999