[For the entire collection see Zbl 0741.00062.]
Let \(H\) be a Hopf algebra over a field \(k\), \(A\) a right \(H\)-comodule algebra with structure map \(\rho\), and \(A^{\text{co}H}\) the algebra of coinvariants, then \(A/A^{\text{co}H}\) is called an \(H\)-Galois extension if the map \(\beta: A\otimes_{A^{\text{co}H}}A\to A\otimes_ k H\) given by: \(a\otimes b\to(a\otimes 1)\rho(b)\) is bijective. The purpose of this very well written note is twofold. First give examples of such extensions among them classical Galois field extensions, smash products and strongly graded algebras. The second purpose is to survey recent results concerning Galois extensions, with emphasis on simple smash products and work on affine quotients. An example of a Hopf algebra \(H\) acting on a field \(A\), so that \(A\# H\), is not simple is given here. This example has been used since for various other purposes.