Hansen-Mullen不可约猜想的一个新证明
A new proof of the Hansen-Mullen irreducibility conjecture
摘要：
给出了Hansen-Mullen不可约性猜想的一个新证明。证明依赖于一个应用程序的（看似新的）充分条件存在的元素的程度$n$在支持功能的有限域。这种与不可约多项式的联系是通过具有有限域中的值的函数的离散傅里叶变换（DFT）的最小周期来实现的。我们利用这种关系，并证明，在一个基本的方式，相关功能的DFT特征基本对称函数（产生系数的特征多项式）有足够大的最小周期（除了一些真正的例外）。这与文献中用于解决具有指定系数的不可约多项式的存在性的先前技术形成鲜明对比。
英文摘要：
We give a new proof of the Hansen-Mullen irreducibility conjecture. The proof relies on an application of a (seemingly new) sufficient condition for the existence of elements of degree $n$ in the support of functions on finite fields. This connection to irreducible polynomials is made via the least period of the discrete Fourier transform (DFT) of functions with values in finite fields. We exploit this relation and prove, in an elementary fashion, that a relevant function related to the DFT of characteristic elementary symmetric functions (which produce the coefficients of characteristic polynomials) has a sufficiently large least period (except for some genuine exceptions). This bears a sharp contrast to previous techniques in literature employed to tackle existence of irreducible polynomials with prescribed coefficients.
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