How Does Hadlock Prove Every Polynomial of Degree n Has a Symmetric Group Sn?

[Ray]

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.

 

[Greg Bernhardt]

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]

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.

 

[Ray]

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.