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 Jan 18 revised Direct proof of: $\#($cover of $V) < \#F \; \Rightarrow \;V$ belongs to cover edited title Jan 18 asked Direct proof of: $\#($cover of $V) < \#F \; \Rightarrow \;V$ belongs to cover Jan 14 revised Why isn't the orbit-stabilizer theorem obvious? added 33 characters in body Jan 14 revised Why isn't the orbit-stabilizer theorem obvious? deleted 26 characters in body Jan 14 revised Why isn't the orbit-stabilizer theorem obvious? added 137 characters in body Jan 14 asked Why isn't the orbit-stabilizer theorem obvious? Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? ...user18921 begins to explain this. Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? @BillDubuque: thanks; those links you posted are instructive. BTW, one reason why I've been somewhat resistant to convenience as a rationale is recalling a professor of algebra who was constantly compelled to add the caveat "assuming the ring is non-trivial" in his lectures. I'm trying to understand why this inconvenience is not enough to warrant disallowing zero rings by definition. Of course, in the end these arguments on the basis of "convenience" devolve into very subjective accountings of how much inconvenience is acceptable, etc. The answer given by... Jan 11 comment When is a vector space (over field $K$) also a ring (with subring $K$)? @KonstantinArdakov: I'd be glad to accept your comments as the answer to my question if you care to post them as such. Jan 11 accepted What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? Jan 11 reviewed Reject Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1 Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? @Karene: the status of the "field with one element" in Mathematics seems to be greater than I'd expect for something whose impossibility is merely a matter of convention. Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? @Pierre-YvesGaillard: please spare me the condescension. I don't know what you're getting at. It looks like, at most, it would entail requiring that the field that enters in the definition of a vector space be non-trivial. Jan 11 revised What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? deleted 5 characters in body Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? @Pierre-YvesGaillard: I've never thought about either, so I have no opinion. Jan 11 comment What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? If that's the rationale, it'd be the weakest rationale for a mathematical definition that I've ever seen. For one thing, the same thing could be said of the trivial ring, but this ring is not ruled out by the definition. Moreover, in mathematical writing it is commonplace to obviate such annoyances with stipulations like "we henceforth assume that all fields under consideration have $1\neq 0$." Jan 11 revised What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? added 311 characters in body Jan 11 asked What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring? Jan 11 comment When is a vector space (over field $K$) also a ring (with subring $K$)? @KonstantinArdakov: Thanks! I can "fill the gaps" (I think) in your description for the case where $V$ is finite dimensional, but not in general. How does one construct the complement of $Ke$ when one cannot assume that $V$ is finite-dimensional? Jan 11 answered Pedagogy: How to cure students of the “law of universal linearity”?