The Stacks project

Let $A'$ be a subring of $A$ should be let $A'$ be a subring of $K$.

Fixed. Thanks!

Is it accurate to assume that in the expression $\sum_{i=0}^d t_i x^i$, $t_i\in\mathfrak{m}$ for all $i$, or do we just know that $t_0\in\mathfrak{m}$? I have only succeeded in proving that second statement (and in fact that second statement is all that is necessary).

Thanks for your question. I have clarified the argument. See here.

This is not a big deal, but strictly speaking, one should say for any non-zero $x\in K$, blabla as you talk about $x^{-1}$. The same issue persists in the next lemma.

OK, I am going to leave this alone, but I would like to defend it. First of all, it is not an issue in the proof. But even in the statement, I think it is OK. Namely the statement says that either $x \in A$ or $x^{-1} \in A$ or both. So the question is whether in the English language the following makes sense: either the sky is blue or I will eat a blue whale or both. I think this does make sense even though I couldn't possibly eat a blue whale! I mean you just stop reading the sentence after the first occurence of blue, right?

Johan, actually the analogy is disputable: "I will eat a blue whale" is a false sentence, but arguably makes sense. On the other hand, "$x^{-1}\in A$" does not make sense, strictly speaking, without the provision that $x\neq0$.