Chow's paper also at https://arxiv.org/abs/math/9805045 On 1/3/19, Fred Lunnon <fred.lunnon@gmail.com> wrote:
eulerphi(p) -> eulerphi(n) , perhaps? WFL
On 1/3/19, Adam P. Goucher <apgoucher@gmx.com> wrote:
'Insoluble', maybe...?
What I find remarkable is that solving arbitrary cubics is sufficient to solve arbitrary quartics. Group-theoretically, this is just because S_4 has no Jordan-Holder factors beyond those of S_3. But it still seems surprising, because it's tempting to define a 'polynomial hierarchy' (sorry!) of algebraic numbers where we define:
Q_d := smallest subfield of C closed under being able to completely solve all degree-d polynomials with coefficients in Q_d
so that Q_1 are the rational numbers, Q_2 are the compass-and-straightedge-constructible numbers, Q_3 are the origami-constructible numbers, and so forth (with the union being the algebraic numbers themselves). But then you get the astonishing:
Q_1 < Q_2 < Q_3 == Q_4 < Q_5 < Q_6 < Q_7 < Q_8 < Q_9 < ...
where there's a single isolated 'collapse' (equality) between Q_3 and Q_4, and strict inclusions elsewhere.
You can compute nth radicals in Q_p and above, where p is the largest prime factor of eulerphi(p).
--------
One of my favourite fields is Timothy Chow's field of EL-numbers (generated by 0 and closed under exp and log). His paper is definitely worth a read:
http://timothychow.net/closedform.pdf
Best wishes,
Adam P. Goucher
Sent: Thursday, January 03, 2019 at 1:11 PM From: "Bill Gosper" <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: [math-fun] Is there a name for the (presumably 100% of) algebraic numbers inexpressible in radicals?
(E.g., the four duodecics @ http://mathworld.wolfram.com/SquareDissection.html). If not, I propose irradical. —rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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