Gen Poly has Sn group Hadlock

Can anyone explain the idea behind Hadlock's proof that there is an Sn for every poly of degree n? Theorem 37 page 217
I can follow how to build up G from F using symmetric functions and the primitive element theorem. A lso I get the idea of constructing a poly of deg n! from one of deg n. But he starts with rationals beta1 etc to make G irreducible and I don't see the connection back down to F.

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

mathwonk
Homework Helper
2020 Award
Your question is a little vague to me, especially since I do not have the book, and Amazon does not allow searching Theorem 37 on page 217. They do show p. 216, lemma 37f, where Hadlock proves irreducibility of F however. What is it about F that you want to know?

Your question is also a bit muddled. Hadlock is apparently proving that for every positive integer n, there is a polynomial of degree n, which is irreducible over Q and has Galois group S(n).

The idea is apparently to find such a polynomial with variable coefficients and then specialize the coefficients carefully so the result remains irreducible.

Hope I can make my question clearer: Why does the first line of Hadlock's proof begin "By Hilbert's irreducibility theorem...."?

ie Hadlock starts with rationals beta1 etc to make G irreducible and I don't see the connection back down to F.