The polynomial \(F_{n}(T)=(T-x_1)\cdots (T-x_n)\), where the \(x_i\) are indeterminants, is called the general polynomial of degree \(n\). The group \(S_{n}\) permutes the \(x_{i}\) and acts as automorphisms of the field \(E(x_1,\ldots, x_n)\) where \(E\) is any field.
The coefficients of \(F_{n}(T)\) are, up to sign, the elementary symmetric functions \(s_{i}\) of the roots \(x_{i}\). Therefore the field \(E(s_1,\ldots, s_n)\) is contained in the fixed field of \(S_{n}\) on \(E(x_1,\ldots, x_n)\). Therefore \([E(x_1,\ldots, x_n):E(s_1,\ldots, s_n)]\ge n!\).
On the other hand, \(E(x_1,\ldots, x_n)\) is the splitting field of \(F_{n}(T)\) over \(E(s_1,\ldots, s_n)\). Therefore \([E(x_1,\ldots, x_n):E(s_1,\ldots, s_n)]\le n!\).
Thus the galois group of this extension is \(S_{n}\).
In particular any symmetric function in the roots of a polynomial can be written in terms of the coefficients of the polynomial.
The discriminant of a polynomial is the product of the differences of its distinct roots squared:
\[ \Delta=\prod_{i<j} (x_{i}-x_{j})^2 \]
It is a symmetric function of the roots.
If \(\Delta\) is a square, then the galois group of the polynomial is contained in the alternating group.
A radical extension \(K/F\) is a field extension that can be constructed by a succession of simple radical extensions where \(K_{i+1}=K_{i}(\alpha_{i+1})\) where \(\alpha_{i+1}^{n_{i+1}}\in K_{i}\).
Notice that any field extension generated by roots of unity is a radical extension.
Theorem (Kummer): Suppose that \(F\) is a field containing the \(n^{th}\) roots of unity where the characteristic of \(F\) does not divide \(n\). Then \(K/F\) is a cylic galois extension (i.e. has cyclic galois group) of degree \(n\) if and only if \(K=F(\alpha)\) where \(\alpha^{n}\in F\).
A polynomial is “solvable by radicals” (meaning its roots like in a radical extension) if and only if the Galois group is solvable.
The polynomial \(x^5-6x+3\) has Galois group \(S_{5}\).