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Examples and Applications

A few additional remarks on Galois Theory.

The “general polynomial”

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

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.

Solvability by radicals

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 $\alpha^{n}\in F$.

Solvability by radicals

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 $S_{5}$.

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