The complex numbers are just elements of R[i]/(i^2+1). I don't even understand how people are able to get this wrong.

FillMaths 5 hours ago | parent | next [-]

Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.

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cperciva 4 hours ago | parent [-]

I mean, yes of course i is an element in C, because it's a monic polynomial in i.

There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.

	▲	FillMaths an hour ago \| parent [-]
		It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.

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5 hours ago | parent | prev [-]

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