I am looking for a credible source (book / online text) containing the statement and proof of Galois Descent for Algebras. From what I gathered, the statement looks like:
Let $K/F$ be a (finite) Galois extension with Galois group $\Gamma := \operatorname{Gal}(K/F)$.
For a $K$-algebra $B$, a semilinear action by $\Gamma$ is a map $\varphi : \Gamma \times B \to B$, denoted $(\sigma, b) \mapsto \sigma(b)$ satisfying:
- $\sigma(b+b') = \sigma(b) + \sigma(b')$
- $\sigma(bb') = \sigma(b)\sigma(b')$
- $\sigma(kb) = \sigma(k) \sigma(b)$
- $(\sigma\tau)(b) = \sigma(\tau(b))$
for every $\sigma, \tau \in \Gamma$ and $k \in K$ and $b, b' \in B$.
Then, the following two categories are equivalent:
- The category of $F$-algebras
- The category of $K$-algebras with semilinear action by $\Gamma$
The forward direction sends $A$ to $A \otimes_F K$ and the backward direction sends $B$ to $B^\Gamma$, the $\Gamma$-invariant elements of $B$.
I am not sure if the statement is correct, since I gathered it from various sources about Galois descent for vector spaces and some guesswork on my part.
There is also this youtube video which omits conditions 1,2,4.
What I'm looking for
- Preferably a short and simple proof. I'm aware that this might be a special case of some more general theory, but I would prefer elementary approach. A short pdf entirely dedicated to this topic would be great.
What I'm not looking for
- I'm not looking for Galois descent for vector spaces. This has already been done by Keith Conrad.
- I do not think that this thread contains the answer I'm looking for.
I know it's 4 years old, but I think what you probably want is actually section 5 of the notes by Conrad you have already mentioned! He treats Galois descent for algebras at the end.
Also, whilst your statement should be true, the statements in the notes by Conrad are stronger - for a fixed $K$-algebra $\mathcal{A}$, it relates $F$-forms of $\mathcal{A}$ with $\Gamma$-structures on $\mathcal{A}$ in quite a precise way.