Let $E/k$ be a separable field extension of degree $n$. Assume $\text{char}(k)\ne 2$. Classically, one may define the discriminant of $E/k$ as follows. Write $E=k(\theta)$ by primitive root theorem and consider a normal closure $N/k$ of $E/k$. Then $N/k$ is a Galois extension whose Galois group $G$ has an embedding into $S_n$ (via its action on conjugates of $\theta$). The field corresponding to $G\cap A_n$ is either a quadratic extension $k(\sqrt{d})$, in which case we define the discriminant to be $d$, or $k$ itself, in which case we define the discriminant to be $1$.
Recently, I've learnt that using Grothendieck Galois theory, we can define the discriminant as follows. Let $E$ be an etale $k$-algebra of degree $n$, split over a Galois extension $N/k$. Let $G=\text{Gal}(N/k)$. Then $E$ corresponds to a homomorphism $G\to S_n$. Composing with the sign map $S_n\to\{\pm1\}\cong S_2$, we get a homomorphism $G\to S_2$ which amounts to a quadratic etale algebra $Q$. Every quadratic algebra over a field $k$ of characteristic $\ne 2$ is of the form $k[x]/(x^2-d)$. We then define this $d$ to be the discriminant.
My questions:
- How can we verify that these two definitions are the same?
- As an example, what would $Q$ be when $E=k[x]/(x^3+ax+b)$? Note that this makes sense even if the characteristic is not 2.