§9
Galois Extensions
The inclusion-reversing correspondence in §8 is not necessarily a bijection between all intermediate extensions and all subgroups. A necessary condition is that the fixed field of \(\mathrm{Gal}(K/F)\) equals \(F\), i.e.\ \(K^{\mathrm{Gal}(K/F)} = F\).
Let \(K\) be an algebraic extension of a field \(F\). We say that \(K/F\) is a Galois extension if the fixed field of the Galois group \(\mathrm{Gal}(K/F)\) is equal to \(F\), that is, we have \[ K^{\mathrm{Gal}(K/F)} = F. \]
Note that any field automorphism \(\sigma\) of \(K\) can be viewed as a group homomorphism from \(K^\times\) to itself, where \(K^\times\) denotes the multiplicative group of non-zero elements of \(K\).
Let \(G\) be a group, and let \(K\) be a field. A character of \(G\) over \(K\) is a group homomorphism from \(G\) to the multiplicative group \(K^\times\) of non-zero elements of \(K\). The set of all characters from \(G\) to \(K\) is denoted by \(\mathrm{Hom}(G, K^\times)\).
Let \(\sigma_1,\ldots,\sigma_n\) be distinct characters of a group \(G\) over a field \(K\). Then the characters \(\sigma_1,\ldots,\sigma_n\) are linearly independent over \(K\): if \[ \sum_{i=1}^n c_i \sigma_i = 0 \] for some \(c_1,\ldots,c_n \in K\), then \(c_1 = c_2 = \cdots = c_n = 0\).
The following result is a consequence of Dedekind's lemma.
Let \(K/F\) be a finite extension. Then the Galois group \(\mathrm{Gal}(K/F)\) is a finite group, and its order is at most the degree of the extension \(K/F\), that is, \[ |\mathrm{Gal}(K/F)| \le [K:F]. \]
Let \(G\) be a finite subgroup of the group of automorphisms of a field \(K\). Then the order of \(G\) is equal to the degree of the extension \(K/K^G\), that is, \[ |G| = [K : K^G]. \] Consequently, the extension \(K/K^G\) is finite and Galois, and has \(G\) as its Galois group, that is, \(G = \mathrm{Gal}(K/K^G)\).
A finite extension \(K/F\) is Galois if and only if the order of the Galois group \(\mathrm{Gal}(K/F)\) is equal to the degree of the extension \(K/F\), that is, \[ |\mathrm{Gal}(K/F)| = [K:F]. \]
Let \(K/F\) be a field extension, and let \(\alpha \in K\) be algebraic over \(F\). Then the order of the Galois group \(\mathrm{Gal}(F(\alpha)/F)\) is equal to the number of distinct roots of the minimal polynomial of \(\alpha\) over \(F\) in \(F(\alpha)\). Consequently, the extension \(F(\alpha)/F\) is Galois if and only if the number of distinct roots of the minimal polynomial of \(\alpha\) over \(F\) in \(F(\alpha)\) is equal to the degree of the extension \(F(\alpha)/F\).
Determine the Galois group of the extension \(\mathbb{Q}(\sqrt[3]{3})/\mathbb{Q}\) and its subgroups, and determine the corresponding intermediate extensions of \(\mathbb{Q}(\sqrt[3]{3})/\mathbb{Q}\).
Similar problem as the previous exercise, but for the extension \(k(t)/k(t^2)\), or for the extension \(k(t)/k(t^p)\) for any prime \(p\) and any field \(k\) of characteristic \(p\).
Show that the only automorphism of \(\mathbb{Q}\) is the identity.
Show that the only automorphism of \(\mathbb{R}\) is the identity.
Let \(F\) be a field of characteristic not equal to \(2\). Let \(K\) be an extension of \(F\) of degree two. Prove that \(K = F(\alpha)\) for some \(\alpha \in K\) with \(\alpha^2 \in F\). Show that \(K\) is Galois over \(F\). What can be said about degree two extensions of a field of characteristic \(2\)?
Show that there exist automorphisms \(\sigma,\tau\) of \(k(x)\) such that \(\sigma(x) = 1/x\) and \(\tau(x) = 1 - 1/x\). Describe the subgroup of \(\mathrm{Gal}(k(x)/k)\) generated by \(\sigma,\tau\).
§11
Normal Extensions
Let \(K\) be an extension of \(F\). Show that \(K\) is a splitting field over \(F\) for a set \(\{f_1,\ldots,f_n\}\) of polynomials in \(F[x]\) if and only if \(K\) is a splitting field over \(F\) for the polynomial \(f_1 \cdots f_n\).
Let \(K\) be a splitting field of a set \(S\) of polynomials over \(F\). If \(L\) is a subfield of \(K\) containing \(F\) for which each \(f \in S\) splits over \(L\), show that \(L = K\).
Provide an example of a field extension \(F \subseteq K \subseteq L\) such that \(L/K\) and \(K/F\) are normal, but \(L/F\) is not normal.
Let \(K\) be an extension of a field \(F\). Assume that \(K\) is algebraically closed. Show that the algebraic closure of \(F\) in \(K\) is an algebraic closure of \(F\).
Provide an example of a field \(K\) such that \(\mathrm{Aut}(K)\) contains an element of infinite order.