Definition of an etale Algebra

Question

I am reading Szamuely´s book "Galois groups and fundamental groups" and I am somewhat confused about how he is defining an etale algebra:

A finite dimensional k-algebra A is etale (over k) if it is isomorphic to a finite direct product of separable extensions of k.

Now I have seen several other definitions regarding an etale algebra, namely it being isomorphic to a finite direct product of FINITE separable extensions of k or even distinguishing etale algebras and finite algebras for this specific point.

For Theorem 1.5.4 he then uses the wording of an finite etale algebra. Does he now refer to the definition with finite separable extensions, because that is what I think is needed for the proof.

Answer 1

I don't think there is much ambiguity going on: you quote "A finite dimensional k-algebra A is étale (over k) if...", so the definition only applies to finite-dimensional algebras, and in particular it has to be a product of finite separable extensions.

And yes, saying "finite étale algebra" is a way to insist on this finite dimensionality. Here it seems redundant given the definition, but in general there are notions of possibly infinite étale algebras, so it can be worth to make it clear that we are only interested in finite stuff.