# Reference for Galois Descent for Algebras

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:

1. $\sigma(b+b') = \sigma(b) + \sigma(b')$
2. $\sigma(bb') = \sigma(b)\sigma(b')$
3. $\sigma(kb) = \sigma(k) \sigma(b)$
4. $(\sigma\tau)(b) = \sigma(\tau(b))$

for every $\sigma, \tau \in \Gamma$ and $k \in K$ and $b, b' \in B$.

Then, the following two categories are equivalent:

1. The category of $F$-algebras
2. 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.