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I've just seen a corollary in P.Etingof's book Introducion to Representation Theory which states that every irreducible finite-dimensional representation of a commutative algebra is 1-dimensional: enter image description here

The similar result I've come across before, characterizing irreducible finite-dimensional representations of abelian groups, requires a representation over an algebraically closed field, while Corollary 2.3.12. seems to apply to any field as shown in the screenshot.

Besides, the author uses Schur's lemma(Corollary 2.3.10.), which requires the base field to be algebraically closed, to prove Corollary 2.3.12. as in the following two pictures:

enter image description here

enter image description here

So I wonder if Corollary 2.3.12 is a stronger version of the proposition characterizing irreducible finite-dimensional representations of abelian groups over an algebraically closed field, or that the statement of this corollary is incomplete? Thanks in advance!

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    $\begingroup$ page 8, second sentence of section 2.2: Unless stated otherwise, we will always assume that $k$ is algebraically closed $\endgroup$
    – David A. Craven
    Commented Feb 17, 2022 at 10:15
  • $\begingroup$ @DavidA.Craven Thanks. I didn't notice this sentence. This is really a stupid question. $\endgroup$
    – zyy
    Commented Feb 17, 2022 at 13:11
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    $\begingroup$ When I'm writing stuff with assumptions like this, I tend to repeat them at the start of each section. It's a very valid point for you to make. Over a non-algebraically closed field, representations can be $2$-dimensional, for example $\mathbb C$ over $\mathbb R$. $\endgroup$
    – David A. Craven
    Commented Feb 17, 2022 at 13:29
  • $\begingroup$ @DavidA.Craven Thank you for your patience. I think I do understand now. $\endgroup$
    – zyy
    Commented Feb 17, 2022 at 14:17

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