Any textbook on Galois theory contains a proof that a polynomial equation with solvable Galois group can be solved by radicals. From a practical point of view, we need to find suitable representations of the group and the roots of the polynomial. We first reduce the problem to that of cyclic extensions of prime degree and then work out the radicals, using the work of Girstmair. We give numerical examples of Abelian and non-Abelian solvable equations and apply the general framework to the construction of Hilbert Class fields of imaginary quadratic fields.
任何有关伽罗瓦理论的教科书都包含一个证明,即具有可解伽罗瓦群的多项式方程可以由根基求解。从实践的角度来看,我们需要找到组的适当表示形式和多项式的根。我们首先将问题简化为素数的周期性扩展,然后使用吉尔斯特梅尔的研究方法求出根部。我们给出了Abelian和非Abelian可解方程的数值示例,并将一般框架应用于虚二次场的Hilbert类场的构建。 P>
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