Does anybody know a reference for basic properties of tensor products of (finite) algebraic extensions of fields?


Ideally, I would like a description of $L \otimes\_k K$ for arbitrary finite extensions $L, K$ of $k$ but I would settle for a reference for results such as


1) If $K / k$ is Galois with group $G$ then $K \otimes\_k K \cong \oplus\_{g \in G} K$.


2) If $K / k$ is purely inseparable then $K \otimes\_k K$ is local with residue field $K$ and length $[K : k]$.


3) If $K / k$, $L / k$ are separable then $K \otimes\_k L$ has no nilpotent elements.



Reference for tensor products of fields
Dear anon, the most complete reference might be Bourbaki's Algèbre, Chapter V.


**For question 1**, I suggest Bourbaki's Algèbre, Chapter V, §10, 4. Descente galoisienne, Corollaire . There the Master proves the more general result that the canonical morphism 
$$K\otimes\_{k} K\to K^G: x\otimes y\mapsto (x \sigma (y))\_{\sigma \in G} $$
is injective for any Galois extension $K/k$, finite *or infinite*, with Galois group $G$, and bijective if the extension is finite.


**Question 3** is trivial from Bourbaki's point of view since for Him the definition of $K/k$ being a separable extension is that for any field extensions $L/k$, the $k$-algebra $K\otimes \_{k} L$ has no nilpotents (neither $K$ nor $L$ is assumed finite-dimensional over $k$). As a concession to less enlightened mortals, He proves in §15, Exemple 3, that if the extension $K/k$ is algebraic ( for example finite-dimensional) this notion coincides with the one that you and I are familiar with: the minimal polynomial of any element in $K$ has simple roots . 


**For question 2**, I cannot give you a reference which exactly answers your question. However a purely inseparable extension is a particular case of a primary extension and these are considered at the end of our reference, in §17,2. Produit d'extensions . The *Corollaire* there shows that the nilpotent radical $P$ of $K\otimes\_{k} K$ is prime and since this algebra is finite-dimensional, it is local of dimension zero with unique prime ideal $P$ . We still must prove that its length is $[K:k]$. This is equivalent to the claim that the $K$-algebra $ (K\otimes\_{k} K) /P $ is $K$ . This follows from the existence of the product map $K\otimes K \to K$ sending $x\otimes y$ to $xy$.The kernel of this map is exactly the unique prime ideal $P$ of $K\otimes K$ . (By the way, an excellent reference for the notion of "length" is Appendix A to Fulton's book *Intersection Theory* ; Example A.1.1 page 407 is relevant to the above discussion)


**PS** If you are not familiar with exotic languages, you will be relieved to know that this volume of Bourbaki exists in English translation. 


