Example 6

Suppose that f is monic of degree > 0 and K is of characteristic ≠ 2. Define on A a total ordering, denoted by ≤, and put δ(f) = $\prod_{\alpha < \beta} (\beta - a)$, $(\alpha, \beta) \in A \times A$, and for each $\sigma \in \mathfrak{S}_A$ put $\delta_\sigma(f) = \prod_{\alpha < \beta} (\sigma(\beta) - \sigma(\alpha))$. We have $\delta_\sigma(f) = \varepsilon(u) \delta(f)$, where $\varepsilon(u)$ is the signature of $\sigma$ (I, p. 64) and $\delta(f) \neq 0$. For every $\tau \in \mathrm{Gal}(N/K)$ we have $\tau(\delta(f)) = \delta_{\tau|_A}(f)$. Hence $\Gamma$ is contained in the alternating group $\mathfrak{A}_A$ if and only if $\delta(f) \in K$. Moreover $\delta(f)^2 = \prod_{\alpha < \beta} (\beta - a)' = d(f)$ is the discriminant of the polynomial f (IV, p. 81). Hence $\Gamma \subset \mathfrak{A}_A$ if and only if $d(f)$ is the square of an element of K. Thus in Example 2 we have $d(f) = 49 = 7^2$ and in Example 3, $d(f) = -108$ (IV, p. 85).

Let N be a Galois extension of K and let L be a subextension of N which is Galois over K. Every K-automorphism $\sigma$ of N induces a K-automorphism $\sigma_L$ of L (V, p. 55, Remark 1). Therefore the mapping $\sigma \mapsto \sigma_L$ is a homomorphism of $\mathrm{Gal}(N/K)$ into $\mathrm{Gal}(L/K)$, called the restriction homomorphism.