[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
'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
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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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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Sent: Thursday, January 03, 2019 at 4:17 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Is there a name for the (presumably 100% of) algebraic numbers inexpressible in radicals?
eulerphi(p) -> eulerphi(n) , perhaps? WFL
Yes, thanks for spotting that! Interestingly, on the subject of Timothy Chow's EL numbers and suchlike, there is an interesting disparity between R and C: 1. The first-order theory of C with ring operations and exp() is undecidable. 2. The first-order theory of R with ring operations and exp() is decidable, assuming Schanuel's Conjecture. The proof of the first of these is quite straightforward; we can define the rationals: c is rational <==> (there exists a : there exists b : (c*b == a) and (exp(a) == 1) and (exp(b) == 1)) and then appeal to Julia Robinson's proof that arithmetic over Q is undecidable. The second of these is mentioned here: https://en.wikipedia.org/wiki/Tarski%27s_exponential_function_problem Best wishes, Adam P. Goucher
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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I dont think there is. are you SURE that 100% of them are inexpressible in radicals???? On Thu, Jan 3, 2019 at 8:12 AM Bill Gosper <billgosper@gmail.com> wrote:
(E.g., the four duodecics @ http://mathworld.wolfram.com/SquareDissection.html). If not, I propose irradical. —rwg
participants (4)
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Adam P. Goucher -
Bill Gosper -
Fred Lunnon -
Gabe K