- Hmm, Wikipedia says
"A function u of a differential extension G of a differential field F is an elementary function over F if it belongs to a finite chain (for inclusion) of differential subfields of G that starts from F and is such that each is generated over the preceding one by a function that is either
algebraic over the preceding field, or
an exponential, that is, ∂ u = u ∂ a for some a ∈ F or
a logarithm, that is, ∂ u = ∂ a / a for some a ∈ F
With this definition, the usual elementary functions are exactly the function that are elementary over the field of the rational functions."
But the "algebraic over the preceding field" seems to claim the solution of any quintic equation counts as an elementary function.
https://en.wikipedia.org/wiki/Elementary_function
John Carlos Baez
npub1nf…3nqe4
2026-04-16 01:02:57 UTC
in reply to nevent1q…0jd3
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2026-04-16 01:02:57 UTCEvent JSON
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"content": "- Hmm, Wikipedia says\n\n\"A function u of a differential extension G of a differential field F is an elementary function over F if it belongs to a finite chain (for inclusion) of differential subfields of G that starts from F and is such that each is generated over the preceding one by a function that is either\n\n algebraic over the preceding field, or\n\n an exponential, that is, ∂ u = u ∂ a for some a ∈ F or\n\n a logarithm, that is, ∂ u = ∂ a / a for some a ∈ F \n\nWith this definition, the usual elementary functions are exactly the function that are elementary over the field of the rational functions.\"\n\nBut the \"algebraic over the preceding field\" seems to claim the solution of any quintic equation counts as an elementary function. \n\nhttps://en.wikipedia.org/wiki/Elementary_function",
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