Over a perfect field, finite normal extensions and polynomial splitting fields are Galois.Suggest a slogan
Let \(k\) be a field.
If \(\ch k = 0\) or \(k\) is finite, then \(k\) is perfect; in particular \(\CC\), \(\RR\), \(\QQ\), and \(\FF_p\) for \(p\) prime are perfect.
If \(k\) is perfect, then every finite normal extension of \(k\) is Galois.
If \(k\) is perfect, then the splitting field over \(k\) of every nonconstant polynomial \(f\in k[x]\) is Galois over \(k\).