Sage not returning roots of polynomimal

edit

This is a further answer to the question, although fidbc and Emmanuel Charpentier did already said all.

But the given polynomial is a special one, it is a reciprocal polynomial in $q^2$, so we may go some steps further. (But unfortunately i could not find an explicit form either.)

So it is natural to make the substitution $$ t = q^2+\frac 1 {q^2}=q^2+r^2\ , r =\frac 1q\ .$$ I am doing this substitutions my way...

R.<q,r, t> = PolynomialRing(QQ)
W = q^32 - q^30 + 3*q^28 - 3*q^26 + 6*q^24 - 6*q^22 + 9*q^20 - 9*q^18 + 12*q^16 - 9*q^14 + 9*q^12 - 6*q^10 + 6*q^8 - 3*q^6 + 3*q^4 - q^2 + 1
J = R.ideal( [ W, q^2 + r^2 - t, q*r-1 ] )
K = J.elimination_ideal( [ q,r ] )
P = K.gens()[0]
print P

and the polynomial in $t$ is after substitution (or rather elimination):

sage: print P
t^8 - t^7 - 5*t^6 + 4*t^5 + 8*t^4 - 5*t^3 - 4*t^2 + t + 2

(The further lines are misleading unless we are searching for exact solutions...)

The roots of this polynomial are:

sage: var( 't' );
sage: P = t^8 - t^7 - 5*t^6 + 4*t^5 + 8*t^4 - 5*t^3 - 4*t^2 + t + 2
sage: for root in P.roots( ring=QQbar, multiplicities=False ):
....:     print root
....:     
1.514032200639550?
1.791264905617066?
-1.480484671919947? - 0.2566820580880113?*I
-1.480484671919947? + 0.2566820580880113?*I
-0.4934011167978715? - 0.4011241207989847?*I
-0.4934011167978715? + 0.4011241207989847?*I
0.8212372355895106? - 0.3652199571071136?*I
0.8212372355895106? + 0.3652199571071136?*I

and the two real $t$-roots correspond to the eight roots of absolute value one of the initial polynomial:

sage: R.<q> = PolynomialRing(QQ)
....: W = q^32 - q^30 + 3*q^28 - 3*q^26 + 6*q^24 - 6*q^22 + 9*q^20 - 9*q^18 + 12*q^16 - 9*q^14 + 9*q^12 - 6*q^10 + 6*q^8 - 3*q^6 + 3*q^4 - q^2 + 1

sage: for w in W.roots( ring=QQbar, multiplicities=False ):
....:     if abs(w) != 1:    continue
....:     print "w = %s |w| = %.2f w^2 + w^-2 = %.8f" % ( w, abs(w), w^2+w^-2 )
....:     
w = -0.9735585377388802? - 0.2284376798948314?*I |w| = 1.00 w^2 + w^-2 = 1.79126491
w = -0.9735585377388802? + 0.2284376798948314?*I |w| = 1.00 w^2 + w^-2 = 1.79126491
w = -0.9372876026918778? - 0.3485569535099145?*I |w| = 1.00 w^2 + w^-2 = 1.51403220
w = -0.9372876026918778? + 0.3485569535099145?*I |w| = 1.00 w^2 + w^-2 = 1.51403220
w = 0.9372876026918778? - 0.3485569535099145?*I |w| = 1.00 w^2 + w^-2 = 1.51403220
w = 0.9372876026918778? + 0.3485569535099145?*I |w| = 1.00 w^2 + w^-2 = 1.51403220
w = 0.9735585377388802? - 0.2284376798948314?*I |w| = 1.00 w^2 + w^-2 = 1.79126491
w = 0.9735585377388802? + 0.2284376798948314?*I |w| = 1.00 w^2 + w^-2 = 1.79126491

So it is enough to understand the roots of $$ P = t^{8} - t^{7} - 5 t^{6} + 4 t^{5} + 8 t^{4} - 5 t^{3} - 4 t^{2} + t + 2\ . $$ Now - sometimes, but only sometimes - there is an implemented luck that allows to factorize a polynomial over some smaller field. In our case there is no obvious choice, since we cannot find easy subfields of the splitting field of $P$.

sage: K.<a> = NumberField( P )
sage: for L, _, _ in K.subfields():
....:     print "Subfield of degree %s and discriminant %s" % ( L.degree(), L.disc().factor() )

Subfield of degree 1 and discriminant 1
Subfield of degree 8 and discriminant -1 * 3^9 * 21157

Now i would need some more mathematical insight about the initial W, this in the hope that we can still solve by radicals. Sometimes, after adjoining a special element there appear subfields, and one can split over them. For instance, starting with the simple polynomial $Q=x^3-x-1$, there are no non-trivial subfields of NumberField(Q). But if we adjoin $\sqrt{-23}$...

sage: _.<x> = QQ[]
sage: Q = x^3 -x -1
sage: K.<b> = NumberField( Q )
sage: for L, _, _ in K.subfields(): print L.degree(), L.disc()
1 1
3 -23
K.<b,c> = NumberField( Q, x^2+23 )
sage: for L, _, _ in K.subfields(): print L.degree(), L.disc()
1 1
2 -23
3 -23
3 -23
3 -23
6 -12167

there is a field of degree $2$, that is helping us to "solve the cubic". Of course, things would be clear, if sage could deliver in real time the Galois closure, for instance...

R.<t> = PolynomialRing( QQ )
P = t^8 - t^7 - 5*t^6 + 4*t^5 + 8*t^4 - 5*t^3 - 4*t^2 + t + 2
K.<a> = NumberField( P )
L.<b> = K.galois_closure()

(Had to abort the computation after a while.) So is there geometric insight for the initial polynomial W. (E.g. do we count points of some variety...?!)

Try :

SQ=w.roots(ring=QQbar)

(or, equivalently, replace CC by QQbar in fidbc's answer). It turns out that :

sage: SQ[0][0].parent()
Algebraic Field

which is exact :

sage: SQ[0][0].parent().is_exact()
True

a property that holds for all the roots thus obtained :

sage: all(map(lambda r:r[0].parent().is_exact(), SQ))
True

(Here, "exact" means that Sage can give you arbitrary precision for a numerical approximation). However, you won't (always : you might get lucky, but there is no guarantee...) get algebraic expressions for that (ISTR that someone called Galois explained why...).

If you need algebraic expressions of these roots and happen to be able to read French, you might have a look at the excellent "Calcul mathématique avec Sage", which IMHO, should be translated in English, and might benefit from an updated edition : Sage(math) has made huge progress since this book was written).

You might also explicitely work in the ring of polynomials in q, but I'll defer to the official documentation...

HTH,