Problem 10

Let $F$ be a field and $f(x)\in F[x]$ be a separable polynomial of degree $n$. Suppose $E$ is the splitting field of $f(x)$ over $F$. Let $\alpha_{1},\alpha_{2},\dots,\alpha_{n}$ be the zeroes of $f(x)$ in $E$. Define

For example, for two distinct zeroes, we have $D=\alpha_{1}-\alpha_{2}$. For three distinct zeroes, we have $D=(\alpha_{1}-\alpha_{2})(\alpha_{1}-\alpha_{3})(\alpha_{2}-\alpha_{3})$. The discriminant of $f(x)$ is defined to be $\Delta=D^{2}$.

1. Verify that the discriminant of $x^{4}-6x^{3}+11x^{2}-6x$ is 144 for the case where $F=\mathbb{Q}$.

2. Prove that $\Delta\in F$.(Hint: consider what an automorphism from $\mathrm{Gal}(E/F)$ does to $D$.)

3. If $\tau\in\mathrm{Gal}(E/F)$ is the transposition of two zeroes of $f(x)$, show that $\tau(D)=-D$.

4. Show that $\sigma\in\mathrm{Gal}(E/F)$ is an even permutation of the zeroes of $f(x)$ if and only if $\sigma(D)=D$.

5. Hence prove that $\mathrm{Gal}(E/F)$ is isomorphic to a subgroup of $\mathrm{A}_{n}$ if and only if $D\in F$.

6. It is known that if $f(x)=x^{3}+px+q$ for some constants $p,q\in F$, then

   $$\Delta=-4p^{3}-27q^{2}.$$

   Hence determine $\mathrm{Gal}(E/F)$ for the case where $F=\mathbb{R}$ and $f(x)=x^{3}+2x-4$.