[GAP Forum] Sqrt for the cyclotomic numbers
Bill Allombert
Bill.Allombert at math.u-bordeaux.fr
Mon Jan 19 21:55:36 GMT 2015
> Le Mardi 20 janvier 2015 2h48, Palcoux Sebastien <sebastienpalcoux at yahoo.fr> a écrit :
>
>
> Hi,
> Is it possible to extend the function Sqrt on the cyclotomic numbers?
> See below what's happen if we try, for example, with the number E(5) :
> gap> Sqrt(E(5));Error, no method found! For debugging hints type ?Recovery from NoMethodFoundError, no 1st choice method found for `Sqrt' on 1 arguments called from<function>( <arguments> ) called from read-eval-loopEntering break read-eval-print loop ...you can 'quit;' to quit to outer loop, oryou can 'return;' to continuebrk>
Sqrt(E(5)) is +/- E(5)^3, and more generally Sqrt(E(p)) is +/- E(p)^((p+1)/2)
for odd p.
gap> (E(5)^3)^2;
E(5)
However, in the general the square root of a cyclotomic number is not a
cyclotomic number, for example Sqrt(2) is a cyclotomic number, but not
Sqrt(Sqrt(2)).
Cheers,
Bill.
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