Abstract: In classical Galois theory over \(\mathbb{Q}\) (char. 0), there is the Galois correspondence matching subfields of an extension with subgroups of the automorphism group. We want to explore the behavior of field extensions in fields of char. \(p>0\), beginning with finite fields \(\mathbb{F}_p\) and moving on to infinite ones, like \(\mathbb{F}_p(t)\). We will focus on when and why does the Galois correspondence fail and showcase some solutions to this problem.