[GAP Forum] Square root of cyclotomic real number

[Alexander Hulpke]

Dear Forum, Dear Dima,

> One way or another, you need functionality to construct and to compute
> in quadratic extensions of cyclotomic fields, and this might be slow,
> unless someone programs a way to do the square root in the cyclotomic
> field (my number theory is not good, but IIRC you would always be able
> to work in cyclotomics, perhaps of bigger degree).

Alas no, and that’s the whole difficulty. The number you construct (as root of n=E(20)-E(20)^9)

> sqrt(-(1/4*I*sqrt(5) + 1/4*sqrt(2*sqrt(5) + 10) - 1/4*I)^7 +
> (1/4*I*sqrt(5) + 1/4*sqrt(2*sqrt(5) + 10) - 1/4*I)^5 - (1/4*I*sqrt(5) +
> 1/4*sqrt(2*sqrt(5) + 10) - 1/4*I)^3 + 1/2*I*sqrt(5) + 1/2*sqrt(2*sqrt(5)
> + 10) - 1/2*I)

has minimal polynomial
x^8-5*x^4+5
with a nonabelian Galois group. By Kronecker-Weber, n has no square root in a cyclotomic field. (Indeed, if every square root of a cyclotomic number was cyclotomic, all 2-groups would be abelian.)

While generic theory will give you a way to express such square roots in radicals, trying to do sensible arithmetic with such iterated radical expressions is rather delicate and will run into surprising problems concerning equality of different expressions (Keyword: Radial Denesting). I have not seen any method to perform such arithmetic universally, reasonably fast, and not prone to potential problems once it comes to issues such as branch cuts.

Best,

   Alexander