MathDB

Let

\alpha_1,\ldots,\alpha_d,\beta_1,\ldots,\beta_e\in\mathbb{C}

be such that the polynomials

f_1(x) =\prod_{i=1}^d(x-\alpha_i)

and

f_2(x) =\prod_{i=1}^e(x-\beta_i)

have integer coefficients.
Suppose that there exist polynomials

g_1, g_2 \in\mathbb{Z}[x]

such that

f_1g_1 +f_2g_2 = 1

.
Prove that \left|\prod_{i=1}^d \prod_{j=1}^e (\alpha_i - \beta_j)\right|=1

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