# Polynomial Integrality and Tensor Products

Define the tensor product of two monic polynomials over a field in the following way: Take $f(x) = \prod_{i = 1}^{n}(x- \alpha_i)$ and $g(x) = \prod_{j = 1}^{m} (x - \beta_j)$, then $(f \otimes g)(x) = \prod_{i,j = 1}^{n,m} (x - \alpha_i \beta_j)$.

Suppose now that we are given two monic polynomials $f(x)$ and $g(x)$ over $\mathbb{C}$ which are integral at a common value $x_{0}$, i.e., $f(x_0), g(x_0) \in \mathbb{Z}$.

Are there necessary and sufficient conditions to ensure that the tensor product $(f \otimes g)(x_0)$ is integral? With what further restrictions on $f$ and $g$ can we assume that such closure holds?

Thanks!

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Of course, this is true for $x_0=0$. I don't see any reason for this to be true for any other value of $x_0$; can you give some examples of what is making you think there should be such a result? – David Speyer Jul 13 '11 at 21:35
@David: well, if $f, g$ both have integer coefficients and $x_0$ is an integer... – Qiaochu Yuan Jul 13 '11 at 22:16
@Qiaochu That's certainly true. Although the right way to phrase that one is "If $f$ and $g$ have integer coefficients, then so does $f \otimes g$." – David Speyer Jul 14 '11 at 12:36