§7
Compositum of Field Extensions
Let \(L_1/F\) and \(L_2/F\) be field extensions, contained in a common extension \(K\) of \(F\). The compositum of \(L_1\) and \(L_2\) over \(F\), denoted by \(L_1 L_2\), is defined to be the smallest subfield of \(K\) that contains both \(L_1\) and \(L_2\). In other words, \(L_1 L_2\) is the intersection of all subfields of \(K\) that contain both \(L_1\) and \(L_2\). The compositum \(L_1 L_2\) is also called the composite field of \(L_1\) and \(L_2\) over \(F\).
If \(L_1/F\) and \(L_2/F\) are finite extensions contained in a common extension \(K\) of \(F\), and if \[ L_1 = F(\alpha_1,\alpha_2,\ldots,\alpha_m), \qquad L_2 = F(\beta_1,\beta_2,\ldots,\beta_n), \] then show that \[ L_1 L_2 = F(\alpha_1,\alpha_2,\ldots,\alpha_m,\beta_1,\beta_2,\ldots,\beta_n). \]
Show that if \(L_1/F\) and \(L_2/F\) are finite extensions contained in a common extension \(K\) of \(F\), then the compositum \(L_1 L_2/F\) is also a finite extension of \(F\), and we have \[ [L_1 L_2 : F] \le [L_1:F]\,[L_2:F]. \tag{1} \] Provide an example of two finite extensions \(L_1/F\) and \(L_2/F\) such that inequality (1) is strict. Moreover, if \([L_1:F]\) and \([L_2:F]\) are coprime, then \[ [L_1 L_2 : F] = [L_1:F]\,[L_2:F]. \]
§8
Automorphisms of Field Extensions
Let \(L/K\) be a field extension. The Galois group \(\mathrm{Gal}(L/K)\) is the set of all field automorphisms of \(L\) that fix \(K\) pointwise, under composition.
Show that \(\mathrm{Gal}(L/K)\) is a group under composition of maps.
Let \(L/K\) be a field extension and \(L = K(X)\). Any \(K\)-automorphism of \(L\) is uniquely determined by its action on \(X\).
Let \(\alpha \in L\) be algebraic over \(K\). Any \(K\)-automorphism \(\sigma\) maps \(\alpha\) to another root of the minimal polynomial of \(\alpha\) over \(K\), and \(\sigma(\alpha)\) has the same minimal polynomial as \(\alpha\).
If \(L/K\) is a finite extension, then \(\mathrm{Gal}(L/K)\) is a finite group.
Determine \(\mathrm{Gal}(L/K)\) for each of the following extensions:
- \(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\)
- \(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}\)
- \(\mathbb{Q}(\sqrt{2},\sqrt[3]{3})/\mathbb{Q}\)
- \(\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}\)
- \(\mathbb{Q}(\sqrt[3]{3})/\mathbb{Q}\)
- \(\mathbb{Q}(i)/\mathbb{Q}\)
- \(\mathbb{C}/\mathbb{R}\)
- \(k(t)/k(t^2)\) where \(k\) is a field
Let \(k\) be a field.
- Determine whether \(k[t]/(t^2+1)\) is a field; if so find its Galois group over \(k\).
- Determine whether \(k[t]/(t^2+t+1)\) is a field; if so find its Galois group over \(k\).
Let \(K/F\) be a field extension, and let \(L\) be an intermediate extension of \(K/F\). Then the Galois group of \(K\) over \(L\) is a subgroup of the Galois group of \(K\) over \(F\).
Let \(H\) be a subgroup of \(\mathrm{Gal}(K/F)\). The fixed field of \(H\) is \[ K^H = \{ x \in K : \sigma(x) = x \text{ for all } \sigma \in H \}. \]
Show that \(K^H\) is an intermediate extension of \(K/F\). For any subset \(S \subseteq \mathrm{Gal}(K/F)\), show \(K^S = K^{\langle S \rangle}\).
So far we have seen that \(\mathrm{Gal}(K/F)\) is a group, that \(K^H\) is an intermediate extension, and that \(\mathrm{Gal}(K/L)\) is a subgroup of \(\mathrm{Gal}(K/F)\) for any intermediate \(L\).
Determine \(\mathrm{Gal}(\mathbb{Q}(\sqrt{2},\sqrt[3]{3})/\mathbb{Q})\), its subgroups, and the corresponding intermediate extensions. Are there more intermediate extensions than subgroups of the Galois group?
Are there similar examples in positive characteristic?
Are there more intermediate extensions than subgroups of the Galois group for \(\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}\)?
Let \(K/F\) be a field extension. Then:
- The map \(L \mapsto \mathrm{Gal}(K/L)\) is inclusion-reversing: \(L_1 \subseteq L_2 \Rightarrow \mathrm{Gal}(K/L_2) \subseteq \mathrm{Gal}(K/L_1)\).
- The map \(H \mapsto K^H\) is inclusion-reversing: \(H_1 \subseteq H_2 \Rightarrow K^{H_1} \supseteq K^{H_2}\).
- For any intermediate \(L\): \(L \subseteq K^{\mathrm{Gal}(K/L)}\). In particular, \(K^H = K^{\mathrm{Gal}(K/K^H)}\) for any subgroup \(H\).
- For any \(S \subseteq \mathrm{Gal}(K/F)\): \(S \subseteq \mathrm{Gal}(K/K^S)\). In particular, \(\mathrm{Gal}(K/L) = \mathrm{Gal}(K/K^{\mathrm{Gal}(K/L)})\) for any intermediate \(L\).
The maps \(L \mapsto \mathrm{Gal}(K/L)\) and \(H \mapsto K^H\) give an inclusion-reversing bijection between intermediate extensions that are fixed fields and subgroups that are Galois groups of \(K\) over some intermediate extension.