Hi,
I'm trying to symbolically calculate the product of all roots of the
polynomial x^5 - 3*x -1.
(The answer should of course be 1).
Numerically I can compute the product as follows:
sage: x = polygen(QQbar)
sage: p=x^5-3*x-1
sage: p.roots(QQbar)
[(-1.214648042698462?, 1), (-0.3347341419433527?, 1),
(1.388791984407255?, 1), (0.08029510011728016? - 1.328355109820654?*I,
1), (0.08029510011728016? + 1.328355109820654?*I, 1)]
sage: prod([i[0] for i in p.roots(QQbar)])
1.000000000000000? + 0.?e-18*I
My first idea was to define the number field:
sage: K.<a> = NumberField(x^5-3*x-1)
and compute the product of all galois conjugates of a
sage: prod(a.galois_conjugates(K.galois_closure('b')))
however, the computation of K.galois_closure('b') never seems to
finish.
Any ideas what I could do?
Thanks,
Philipp
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