论文标题
在与质量的一般线性群有关的线性多项式的galois组上
On Galois groups of linearized polynomials related to the general linear group of prime degree
论文作者
论文摘要
令$ l（x）$为任何$ q $ linearized多项式，$ \ mathbb {f} _q $，$ q^n $。我们考虑$ l（x）+tx $ over $ \ mathbb {f} _q（t）$的Galois组，其中$ t $超过$ \ mathbb {f} _q $。我们证明，当$ n $是Prime时，Galois组始终为$ gl（n，q）$，除非$ l（x）= x^{q^n} $。同等地，我们证明了$ l（x）/x $的算术单曲集群为$ gl（n，q）$。
Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\mathbb{F}_q(t)$, where $t$ is transcendental over $\mathbb{F}_q$. We prove that when $n$ is a prime, the Galois group is always $GL(n,q)$, except when $L(x)=x^{q^n}$. Equivalently, we prove that the arithmetic monodromy group of $L(x)/x$ is $GL(n,q)$.
