论文标题
某些八八个多项式的galois组
Galois groups of certain even octic polynomials
论文作者
论文摘要
令$ f（x）= x^8+ax^4+b \ in \ mathbb {q} [x] $是不可介绍的多项式，其中$ b $是正方形。我们提供了一种方法，该方法完全描述了$ f（x）$的线性分解的分数化模式，并使用$ a $ a $ a和$ b $上的简单算术条件。结果，我们确定了$ f（x）$的确切六个可能的galois组，并将所有这些组完全分类。作为一个应用程序，我们表征了不可约多项式的Galois组$ x^8+ax^4+1 \ in \ mathbb {q} [x] $。我们还使用类似的方法来获得不可约多项式的Galois组$ x^8+ax^6+bx^4+ax^2+1 \ in \ mathbb {q} [x] $。
Let $f(x)=x^8+ax^4+b \in \mathbb{Q}[x]$ be an irreducible polynomial where $b$ is a square. We give a method that completely describes the factorization patterns of a linear resolvent of $f(x)$ using simple arithmetic conditions on $a$ and $b$. As a result, we determine the exact six possible Galois groups of $f(x)$ and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials $x^8+ax^4+1 \in \mathbb{Q}[x]$. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials $x^8+ax^6+bx^4+ax^2+1 \in \mathbb{Q}[x]$.
