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At Galois theory we have the following:
If is not the zero ring and is free with basis , then the cardinality only depends on , and not on the choice of basis
is this right, considering the discussion at the thread on rank?
Only if is assumed to be commutative or noetherian or something, which doesn’t seem to be the case.
I think that I’ve edited the page so that its claims about rank are correct. These claims are far from complete, but they cover what seems to be used (and the complete version is at rank).
Thanks for that.
I’ll check later today with the lecture notes. I think we need to assume commutative rings pretty much throughout. But let me check
Commutative is enough ?
I wrote:
I’ll check later today with the lecture notes.
Ah, silly me, I thought I’d had to go home to see my hardcopy of Lenstra’s lecture there, but we have the online copy
All rings in his convention are commutative and unital. I added the adjective “commutative” to where the first ring that is not a field appears at Galois theory. What is typed there is pretty much directly from Lenstra’s notes. So I think it should be right now. But you can all check.
OK, thanks.
Commutative is enough ?
According to Wikipedia, yes, although there is no citation or proof.
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