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Let $F$ be a field. By a Galois algebra over $F$ I mean a finite etale extension, that is, a product $K = K_1 \times \cdots \times K_r$ of finite (separable) field extensions, of total degree $[K : F] = n$, equipped with a subgroup $G \subseteq \operatorname{Aut}_F K$ of $n$ linearly independent automorphisms. For example, $\mathbb{C} \times \mathbb{C}$ equipped with the set of four automorphisms $(z,w) \mapsto \{(z,w), (\bar{w},z), (\bar{z},\bar{w}), (w,\bar{z})\}$ is a cyclic Galois algebra over $\mathbb{R}$ of degree $4$.

I'd like to find a reference (or counterexample) to the following simple assertions:

  1. $K$ is a Galois algebra iff it is a product $K = K_1^r$ of copies of a field that is Galois over $F$, and the $G$-action permutes the $n$ ring maps from $K$ to $K_1$ simply transitively;
  2. If $H \leq G$ is a subgroup, then $K/K^H$ is also Galois under the natural $H$-action (more precisely, for each field factor of $K^H$, the portion of $K$ above it is Galois under the $H$-action);
  3. Also, if $H$ is normal, then $K^H/F$ is Galois under the natural $G/H$-action;
  4. If $E$ is any extension field of $F$, then $L \otimes E$ is Galois over $E$, with the $G$-action extended linearly;
  5. If $K$ and $K'$ are Galois algebras over $F$ with Galois groups $G$ and $G'$ respectively, then $K \otimes K'$ is Galois under the natural action of $G \times G'$.

A construction due to Bhargava embeds any finite etale extension of degree $n$ (in characteristic $0$) into an $S_n$-Galois algebra.

It seems that the theory of such algebras is similar in depth to, but not a trivial consequence of, Galois theory of fields (e.g. for part 2, how do we compute $[K^H : F]$?) We no longer have that every subalgebra of $K$ is the fixed field of some subgroup; but on the other hand, the elegant properties 4 and 5 are missing from the classical presentation (because the tensor product of fields need not be a field).

It could be that this is all a special case of Grothendieck's theory of finite etale covers of schemes, but I wouldn't like to dredge up Grothendieckian formalism while discussing mainly about elementary properties of number fields, with a bit of class field theory.

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    $\begingroup$ The notion you're missing is that of a $G$-torsor. For a finite group $G$ and ring $K$, an injective finite etale map $K \to E$ (concrete for $K$ a finite product of fields) is a $G$-torsor if $E$ is equipped with a $G$-action over $K$ so $G\times {\rm{Spec}}(E) \to {\rm{Spec}}(E)\times_{{\rm{Spec}}(K)} {\rm{Spec}}(E)$ defined by $(g,x)\mapsto (x, g.x)$ is an isomorphism, or equivalently $E \otimes_K E \to \prod_{g \in G} E$ defined by $a\otimes b\mapsto (ag(b))_g$ is an isomorphism. This gives geometric insight leading to quick easy proofs of 1--5 when $K$ is a finite product of fields. $\endgroup$ – nfdc23 Mar 3 '17 at 3:57
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    $\begingroup$ Your comment about tensor products of fields not generally being fields is precisely why the geometric language is more convenient than the algebraic one: it is easier to visualize keeping track of connected components than primitive idempotents. And yes, this is all a special case of the theory of finite etale covers of schemes. A real virtue of the geometric language is that the analogous constructions and facts for possibly disconnected finite-degree covering spaces of a connected topological space $X$ (with other nice properties if you wish) provide a very vivid way to "see" what is true. $\endgroup$ – nfdc23 Mar 3 '17 at 6:00
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    $\begingroup$ Is Bhargava's construction something other than assigning to any degree-$n$ finite etale cover $X \rightarrow Y$ the $S_n$-torsor $\underline{\rm{Isom}}_Y({\rm{Aut}}_{X/Y}, (S_n)_Y)$ (i.e., the Isom-scheme over $Y$ between two finite etale $Y$-groups of order $n!$, namely the Aut-scheme of $X$ over $Y$ and the constant $Y$-group $S_n$)? $\endgroup$ – nfdc23 Mar 3 '17 at 9:10
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    $\begingroup$ Thank you. I was considering changing the definition to one very similar to nfdc23's first comment, but I missed the connection with torsors. $\endgroup$ – Evan O'Dorney Mar 4 '17 at 17:43
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    $\begingroup$ @nfdc23 Bhargava's construction is defined for arbitrary finite covers and not just etale ones. In the etale case, it is equivalent to what you state, but it provides a good generalization of this to arbitrary covers. $\endgroup$ – Will Sawin Jun 7 '17 at 0:59
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Let $K$ be a separable algebraic closure of $k$. Then the functor sending an etale $k$-algebra $A$ to $Hom(A,K)$ is an equivalence to the category of finite sets with a continuous action of the Galois group of $k$ (baby form of Grothendieck's theory). There is an elementary exposition of this in Milne's notes on Fields and Galois theory. I think you can answer all your questions by looking at the sets with Galois action.

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