Galois Theory, 5th Edition

Chapter 18The General Polynomial Equation

DOI: 10.1201/9781003213949-18

As we saw in Chapter 8, the so-called ‘general’ polynomial is in fact very special. It is a polynomial whose coefficients do not satisfy any algebraic relations. This property makes it in some respects simpler to work with than, say, a polynomial over $\mathbb{Q}$, and in particular it is easier to calculate its Galois group. As a result, we can show that the general quintic polynomial is not soluble by radicals without assuming as much group theory as we did in Chapter 15, and without having to prove the Theorem on Natural Irrationalities, Theorem 8.15.

Chapter 15 makes it clear that the Galois group of the general polynomial of degree n should be the whole symmetric group ${\mathbb{S}}_n$, and ...