论文标题
根据其系数，多项式在有限场上的正态性的标准
A Criterion for the Normality of Polynomials over Finite Fields Based on Their Coefficients
论文作者
论文摘要
$ \ bbb f_q $上的不可约多项式在$ \ bbb f_q $上是正常的，如果其根在$ \ bbb f_q $上线性独立。我们表明，有一个多项式$ h_n（x_1，\ dots，x_n）\ in \ bbb z [x_1，\ dots，x_n] $，独立于$ q $，这样，如果不可差的polynomial $ f = x^n+a_1x^n+a_1x^n q^n q^n q {n q}+cdots+cdots+a__n \ bb f \ bb in \ bb in \ bb in \ bb in \ bb in $ h_n（a_1，\ dots，a_n）\ ne 0 $，然后$ f $在$ \ bbb f_q $上是正常的。多项式$ h_n（x_1，\ dots，x_n）$是针对$ n \ le 5 $明确计算的，部分则以$ n = 6 $计算。当$ \ text {char} \，\ bbb f_q = p $时，我们还表明，有一个多项式$ h_ {p，n}（x_1，x_1，\ dots，x_n）\ in \ bbb f_p [x_1，\ dots，x_n] $，具体取决于$ p $，属于$ p $ huar_n。这些结果对于通过环状galois组在任意场上的可分离性不可隔离多项式的有效。
An irreducible polynomial over $\Bbb F_q$ is said to be normal over $\Bbb F_q$ if its roots are linearly independent over $\Bbb F_q$. We show that there is a polynomial $h_n(X_1,\dots,X_n)\in\Bbb Z[X_1,\dots,X_n]$, independent of $q$, such that if an irreducible polynomial $f=X^n+a_1X^{n-1}+\cdots+a_n\in\Bbb F_q[X]$ is such that $h_n(a_1,\dots,a_n)\ne 0$, then $f$ is normal over $\Bbb F_q$. The polynomial $h_n(X_1,\dots,X_n)$ is computed explicitly for $n\le 5$ and partially for $n=6$. When $\text{char}\,\Bbb F_q=p$, we also show that there is a polynomial $h_{p,n}(X_1,\dots,X_n)\in\Bbb F_p[X_1,\dots,X_n]$, depending on $p$, which is simpler than $h_n$ but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.
