I have a family of polynomials and I want to consider special members of this family. In other words I'm considering polynomials in a ring $R = K[x]$ where $K = \mathbb{Q}[t]$. In sage I do the following:
K = PolynomialRing(QQ, ["t"])
R = PolynomialRing(K, ["x"])
t = K.gen(0)
x = R.gen(0)
f = (t**2 - QQ(1/10)*t + 1)*x**2 + (QQ(3/4)*t + QQ(7/2))*x - t + 8
f1 = f.specialization({t: 1})
This works fine and as expected $f_1$ is a polynomial only in $x$:
f1.parent() == QQ["x"] # True
Now I want to do exactly the same but over $\overline{\mathbb{Q}}$ instead:
L = PolynomialRing(QQbar, ["t"])
S = PolynomialRing(L, ["x"])
t = L.gen(0)
x = S.gen(0)
g = (t**2 - QQ(1/10)*t + 1)*x**2 + (QQ(3/4)*t + QQ(7/2))*x - t + 8
g1 = g.specialization({t: 1})
I would expect $g_1$ to be a polynomial only in $x$ as above, i.e. I would expect $g_1 \in \overline{\mathbb{Q}}[x]$. However, I get:
g1.parent() == QQbar["x"] # False
g1.parent() == S # True
Is this a bug? Or am I misunderstanding something?
