论文标题
关于与特殊线性质量的线性多项式的Galois组
On Galois groups of linearized polynomials related to the special linear group of prime degree
论文作者
论文摘要
让$ f $是主要特征$ p $的字段，让$ q $是$ p $的功率。我们假设$ f $包含订单$ q $的有限字段。 $ q $ -polynomial $ l $ a $ f $ a $ f $ a $ f $是多项式环$ f [x] $的一个元素，其属性是$ x $的$ x $，其符合$ l $的$ l $，而非零系数具有指数$ q $的功率。如果$ l $的指数为$ q^n $，我们说$ l $具有$ q $ -Degree $ n $。我们假设$ l $的$ x $项的系数为非零。我们研究了$ l（x）/x $在$ f [x] $中不可记论的假设，我们调查了$ f $ $ l $ $ l $ $ l $的Galois Group $ g $。众所周知，如果$ l $具有$ q $ -Degree $ n $，其根部是$ n $二维的矢量空间，而订单$ q $和$ g $在此空间上是线性的。 我们的主要定理是以下内容。我们认为，$ Q $ l $ $ Q $ -Degree是Prime，$ r $，例如，$ L（x）/x $不可$ f $以上。假设$ l $的$ x $项的系数是$（-1）^r $，我们表明$ l $ $ f $ over $ f $的Galois组必须是特殊的线性$ sl（r，q）$ r $ y $ r $，$ q $ q $ q $ q $ q> 2 $。我们还证明了投影特殊线性组$ psl（R，Q）$的投影版本，当$ q = 2 $时，我们会介绍分析。
Let $F$ be a field of prime characteristic $p$ and let $q$ be a power of $p$. We assume that $F$ contains the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that those powers of $x$ that occur as terms of $L$ with nonzero coefficient have exponent a power of $q$. If the exponent of the leading term of $L$ is $q^n$, we say that $L$ has $q$-degree $n$. We assume that the coefficient of the $x$ term of $L$ is nonzero. We investigate the Galois group $G$, say, of $L$ over $F$, under the assumption that $L(x)/x$ is irreducible in $F[x]$. It is well known that if $L$ has $q$-degree $n$, its roots are an $n$-dimensional vector space over the field of order $q$ and $G$ acts linearly on this space. Our main theorem is the following. We consider a monic $q$-polynomial $L$ whose $q$-degree is a prime, $r$, say, with $L(x)/x$ irreducible over $F$. Assuming that the coefficient of the $x$ term of $L$ is $(-1)^r$, we show that the Galois group of $L$ over $F$ must be the special linear group $SL(r,q)$ of degree $r$ over the field of order $q$ when $q>2$. We also prove a projective version for the projective special linear group $PSL(r,q)$, and we present the analysis when $q=2$.
