Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by CXF, curve-excluding fields. Curve-excluding fields provide examples to various questions. Model theoretically, they are model complete and algebraically bounded. Field theoretically, they are not large. This answers a question of Junker and Macintyre negatively. Joint work with Will Johnson and Erik Walsberg.