$x^n-1\in \mathbb F_q[x]$的显式分解
Explicit factorization of $x^n-1\in \mathbb F_q[x]$
摘要：
设$\mathbb F_q$是有限域，$n$是正整数.本文证明了在$q$和$n$上的一定条件下，多项式$x^n-1$可分解为不可约二项式$x^t-a$，并给出了分解为不可约因子的显式因式分解.最后，通过削弱我们的假设，我们得到了$x^{2t}-ax^t+b$形式的因子，并给出了$x^n-1$分解为不可约因子的显式分解.
英文摘要：
Let $\mathbb F_q$ be a finite field and $n$ a positive integer. In this article, we prove that, under some conditions on $q$ and $n$, the polynomial $x^n-1$ can be split into irreducible binomials $x^t-a$ and an explicit factorization into irreducible factors is given. Finally, weakening one of our hypothesis, we also obtain factors of the form $x^{2t}-ax^t+b$ and explicit splitting of $x^n-1$ into irreducible factors is given.
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