We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.

中文翻译：

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大次数随机多项式的不可约性

我们考虑具有固定定律的独立同分布系数的随机多项式。假设 Dedekind zeta 函数的黎曼假设，我们证明这样的多项式是不可约的，并且它们的伽罗瓦群包含随着次数趋于无穷大的高概率交替群。这解决了 Odlyzko 和 Poonen 对 Dedekind zeta 函数的 RH 有条件的猜想。