Now for some bookkeeping.
Following a cursory comparison of the three formulae given at
>>7614713 ,https://upload.wikimedia.org/wikipedia/commons/9/99/Quartic_Formula.svg , and
http://planetmath.org/QuarticFormulaIt appears at-a-glance that the latter two formulations are verbatim copies of each other, and the large-scale layout of terms and pm-branching agree with my ending derivation. The latter formulae, however, correspond to the general /MONIC/ quartic (starting with "a" = 1), and using a,b,c,d on down, which is cosmetically different from my presentation, where I insisted upon carrying the leading coefficient through.
So, once again, take care not to confuse your a,b,c's, depending on context. Here, they are distinct from a layout that I would personally prefer (and that I have used in my own presentation), and they are also not to be confused with my presentation of the cubic:
>>7615628 although after scanning a bit, you can start to make out similarities in the forms; something that "looks" very much like our version of the cubic formula is pervasive throughout, as our own y appears not once, not twice, but /three/ times in the derivation, which is why longcat is so very, very long.
Finally, the planetmath link doesn't seem to account for the case when 2y+p = q = 0, which is rather more merciful since y becomes superfluous in this fifth case. You really need this fifth case to call the thing comprehensively solved in terms of an algebraic solution.
Oh yes. I already shouted "FTA, LOL!", but more formally, to the above PROPOSITION: "But this is precisely an algebraic solution of the polynomial equation which was given, being what was to be shown, QED etc etc.
I may amuse myself with a re-write and post of the general formulae in my own terms ITT, but the substantive math is done, apart from two things: complex coefficients (issues with root extraction?), and I still can't get the above Resolvent Cubic thing to verify in any way shape or form.
