I am trying to understand what the term axiomatization exactly means . We didn't really discuss this in the lectures at my school but I have to write a homework related to that and I don't understand the question of the task. Can someone explain what the concept of axiomatization really means? My task is: Given a signature σ1 and a set of formulas Φ1 ⊆ FO[σ1] I have to show that Φ1 "axiomatizate" the class of fields . Of course I don't expect someone to solve the problem for me but I would really appreciate some easy examples of which steps should I follow to solve it (example of how to show the same thing not for fields but for something else) and what exactly this axiomatization concept means.
Thanks in advance!