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Dear Mariano, I can't characterize the extensions in your interesting question, but here is a class of examples. Take for $k$ an algebraically closed field and for $K$ any algebraically closed extens …

(As an aside observe that this is one of the rare cases where the tensor product of two fields is a field) …

Dear Zev, as LASER pointed out, Lang's Proposition 6.11 indeed solves your problem. Still, I'd like to add two remarks that might put your problem in perspective. A) Given a field $K$ and a group of …

Any decent course on field theory will state that in characteristic $p$ an extension of fields $k\subset K$ canonically decomposes as the tower $k\subset K_{sep}\subset K$ with $K$ purely inseparable … [My question was the case $F=\mathbb F_p$] Core result Let $F\subset E$ be an extension of fields with $F$ perfect and $trdeg_{F}(E)=1$. …

Dear Albert, the key theoretical tool in your problem is the theorem (due independently to Auslander-Brumer and Fadeev) relating the Brauer group of a field $k$ and that of its rational function field …

In an answer to this question I was led to show the trick proving that $\mathbb R$ is the fraction field of some strict subring $A\subsetneq \mathbb R=\operatorname{Frac}(A)$. A crucial point in the …

Consider the inclusion $k\subset A$ of the field $k$ in the domain $A$ and the fraction field $K=Frac(A)$ of $A$. Obviously if a family $(a_i)_{i\in I}$ of elements $a_i \in A$ is algebraically indep …

Consider two extension fields $K/k, L/k$ of a field $k$. A frequent question is whether the tensor product ring $K\otimes_k L$ is a field. … An example where we do obtain a field is when the extension fields $K,L$ are finite dimensional over $k$ with relatively prime dimensions. …

Let $k\subset K$ be a completely arbitrary extension of fields. This extension is said to be separable if equivalently a) For all extensions $k\subset L$, the ring $K \otimes _k L$ is reduced. …