– Did you lear about ultrafilters for the first time in a topological context? :-) ]]>

For example 3 ∈ 17 clearly works combinatorically as well as in Finsler’s “generalised numbers”. It’s amusing (at least for me) to try to think of a way to think of a way to do something “backwards” — perhaps in the sense that one looks for a field with one element.

The notion of a prime rectangle could be related to some ambient context perhaps, or some relationship between side-lengths and areas.

Anyway I do agree with you; type errors are a good tool for teachers to identify when a student totally missed the boat.

]]>While the image of the det homomorphism is an integral domain, its domain is not! You can have a non-zero non-invertible matrix $B$ and then $\det(B)=0$ and you can’t say anything about $\det(X)-\det(Y)$. So the proof is valid only when both $A$ and $B$ are invertible (or at least when $A$ and $B$ have a non-zero determinant).

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