Jacobson Lemma from Herstein

Asked Aug 11 '20 at 11:49

Active Aug 11 '20 at 11:54

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This question is asked in chapter-6.8 of Topics in Algebra by I N Herstein -

Prove the Jacobson lemma for elements $A,B\in F_{n}$ if $n$ is less than the characteristic of $F$.

$F_{n}$ is the set of all $n \times n$ matrices.

Jacobson lemma- https://core.ac.uk/download/pdf/82031256.pdf

In lemma, characteristic of $F$ is $0$. So I am confused how to prove this question. What is the meaning of $n$ is less than the characteristic of $F$?

I know the proof when characteristic of $F$ is $0$. It is given in the book.

edited Aug 11 '20 at 11:54

asked Aug 11 '20 at 11:49

Tanya Rastogi

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Please add more details. – General Grievous Aug 11 '20 at 11:52
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The usual proof using derivation should work without modification because $n!\ne0$ when $n<\operatorname{char}(F)$. – user1551 Aug 11 '20 at 15:43
I understand your point. But, if I use this lemma: "If characteristic of $F$ is 0 and tr$A^{i}=0$ for $i≥1$ then $A$ is nilpotent" then what is the meaning of $n$ is less than characteristic of $F$? – Tanya Rastogi Aug 11 '20 at 17:29
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@TanyaRastogi No, it doesn't mean that, but the statement is still true when $n<\operatorname{char}(F)$. See this answer for instance. The conventional proof that uses Newton's identities also work, (again) because $n!\ne0$. – user1551 Aug 12 '20 at 06:03

Jacobson Lemma from Herstein

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