Dear forum,
To complement Dima's answer, if you are using Sage,
you should be aware that Sage provides two useful
exact fields:
- QQbar, the algebraic closure of QQ,
- AA, the "algebraic real field", which is the intersection
of QQbar with the reals.
I would advise you to work in AA for your problem.
Defining, as Dima suggested,
sage: a = libgap.eval('E(20)-E(20)^9').sage()
you get an element in a cyclotomic field:
sage: a
-zeta20^7 + zeta20^5 - zeta20^3 + 2*zeta20
sage: a.parent()
Cyclotomic Field of order 20 and degree 8
Now if you take its square root naively, you end up
in Sage's "Symbolic Ring" which is something to avoid
in general.
sage: b = a.sqrt()
sage: b
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)
sage: b.parent()
Symbolic Ring
The best is to work with AA
sage: AA
Algebraic Real Field
by moving there explicitly
sage: aa = AA(a)
sage: aa
1.902113032590308?
sage: aa.parent()
Algebraic Real Field
Elements of AA are exact, and well-defined:
sage: aa.minpoly()
x^4 - 5*x^2 + 5
Taking square roots stays in AA
sage: bb = aa.sqrt()
sage: bb
1.379171139703231?
sage: bb.parent()
Algebraic Real Field
sage: bb.minpoly()
x^8 - 5*x^4 + 5
I hope this helps.
Samuel
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