
Chapter 18
The General Polynomial Equation
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 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 ...