We present a shortcut to the familiar characterizations of finite Galois extensions, based on an idea from an earlier MONTHLY note by Sonn and Zassenhaus. Let L K be a field extension and G :D Aut.L; K/. We assume throughout that (L : K )<1. Then, automatically,jGj<1. The following result is an essential part of the usual development (and teaching) of Galois theory. Theorem 1. The following conditions are equivalent. (a) jGjD (L : K ). (b) L is a splitting field for a polynomial 06D f 2 K (X ), which does not have multiple roots in L. (c) KDfy2 Lj. y/D y for all 2 Gg. If these conditions hold, then L K is called a Galois extension. For example, condition (b) yields a simple criterion for an extension to be a Galois extension, and condition (c) is crucial for many applications of Galois theory. Combining all three immediately shows that, if L K is a Galois extension and M is an intermediate field, then L M is also a Galois extension, and so MDfy2 Lj. y/D y for all 2 Hg where H :D Aut.L; M/ G;
['Meinolf Geck']
On the characterization of Galois extensions