I am currently trying to read Applications algébriques de la cohomologie de groupes. II: théorie des algèbres simples by J-P. Serre. It is very hard for me to read this article since I'm not a native Frech speaker. Also normal traslation machines can only do so much for you if you need proper mathematical (scientific) texts translated. Now I don't think that MSE is the right place to constantly ask for translations, but I figured someone here might know a good site for translating math articles. This would really help me a lot.

N.B. I don't need a site/person that/who can translate entire articles for me, just small phrases/theorems.

In particular I need the translation of Corollaire 2.


The original text is:

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This translates to:

Corollary 2. Let $L$ be a commutative subfield of an algebra $A$, which is simple, finite and central over $k$. Then $L$ is its own commutant if and only if $[A:k]=[L:k]^2$ or $L$ is the maximal commutative subring of $A$.

Here the commutant $L'$ of $L$ probably means the set of elements of $A$ that commute with every element of $L$. See this. Now the proof:

Let $L'$ be the commutant of $L$ in $A$; since $L$ is commutative, $L'\supseteq L$. From theorem 9, it is clear that $L'=L$ is equivalent to: $$ [A:k]=[L:k]^2 $$ On the other hand, if $L'=L$, every commutative subring of $A$ containing $L$ is inside $L'$, and so is equal to $L$, and $L$ is the maximal commutative subring of $A$. Conversely, if $L$ is so, every element commuting with $L$ is in $L$, and $L'=L$.


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