Chapter 32

Throughout this set of questions, let \(K\) be a root field over \(F\), let \(\mathbf{G}=\operatorname{Gal}(K: F)\) and let \(I\) be any intermediate field. Prove the following: 1 \(I^{*}=G a l(K: I)\) is a subgroup of \(\mathbf{G}\)

The Group of Automorphisms of \(\mathbb{C}\) Prove the following: 1 The only automorphism of \(\mathbb{Q}\) is the identity function. [HINT: If \(h\) is an automorphism, \(h(1)=1\), hence \(h(2)=2\), and so on. \(]\)

Let \(F\) be a field, and \(K\) a finite extension of \(F\). Prove the following : 1 If an automorphism \(h\) of \(K\) fixes \(F\) and \(a\), then \(h\) fixes \(F(a)\).

A Galois Group Isomorphic to \(S_{5}\) Let \(a(x)=x^{5}-4 x^{4}+2 x+2 \in \mathbb{Q}[x]\), and let \(r_{1}, \ldots, r_{5}\) be the roots of \(a(x)\) in \(\mathbb{C}\). Let \(K=\mathbb{Q}\left(r_{1}, \ldots, r_{5}\right)\) be the root field of \(a(x)\) over \(\mathbb{Q} .\) Prove the following \(:\) 1 \(a(x)\) is irreducible in \(\mathbb{Q}\lceil x\rceil\)

. A Cyclic Galois Group 1 Describe the root field \(K\) of \(x^{7}-1\) over \(\mathbb{Q} .\) Explain why \([K: \mathbb{Q}]=6\)

A Galois Group Equal to \(S_{3} .\) 1 Show that \(\mathbb{Q}(\sqrt[3]{2}, i \sqrt{3})\) is the root field of \(x^{3}-2\) over \(\mathbb{Q}\), where \(\sqrt[3]{2}\) designates the real cube root of 2. (HINT: Compute the complex cube roots of unity.)

Computing a Galois Group of Eight Elements Show that \(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})\) is the root field of \(\left(x^{2}-2\right)\left(x^{2}-3\right)\left(x^{2}-5\right)\) over \(Q\).

Computing a Galois Group 1 Show that \(Q(i, \sqrt{2})\) is the root field of \(\left(x^{2}+1\right)\left(x^{2}-2\right)\) over \(\mathbb{Q}\).

Show that \([\mathbb{Q}(\sqrt[3]{2}): \mathbb{Q}]=3\)

Show that the degree of \(Q(\sqrt{2}, \sqrt{3}, \sqrt{5})\) over \(\mathbb{Q}\) is 8 .

Find the degree of \(\mathbb{Q}(i, \sqrt{2})\) over \(\mathbb{Q}\).

Aside from the identity function, there are no Q-fixing automorphisms of \(\mathbb{Q}(\sqrt[3]{2})\). [HINT: Note that \(\mathbb{Q}(\sqrt[3]{2})\) contains only real numbers.]

Explain why \(x^{2}+3\) is irreducible over \(\mathbb{( \sqrt [ 3 ] { 2 } )}\), then show that \([Q(\sqrt[3]{2}, i \sqrt{3}):\) \(Q(\sqrt[3]{2})]=2 .\) Conclude that \([\mathbb{Q}(\sqrt[3]{2}, i \sqrt{3}): \mathbb{Q}]=6\)

List the elements of \(\operatorname{Gal}(\mathbb{Q}(i, \sqrt{2}): \mathbb{Q})\) and exhibit its table.

Write the inclusion diagram for the subgroups of \(\operatorname{Gal}(\mathbb{Q}(i, \sqrt{2}): \mathbb{Q})\), and the inclusion diagram for the fields intermediate between \(\mathbb{Q}\) and \(\mathbb{Q}(i, \sqrt{2})\). Indicate the Galois correspondence.

List the elements of \(\operatorname{Gal}(\mathbb{C}: \mathbb{R})\).

If \(h \in \operatorname{Gal}(\mathbb{Q}(\omega): \mathbb{Q})\), then \(h(\omega)=\omega^{k}\) for some \(k\) where \(1 \leq k \leq p-1\).

Describe the root field \(L\) of \(x^{6}-1\) over \(\mathbb{Q}\), and show that \([L: \mathbb{Q}]=3\). Explain why it follows that there are no intermediate fields between \(\mathbb{\text { and }} L\) (except for \(\mathbb{Q}\) and \(L\) themselves).

Prove that the identity function and the function \(a+b i \rightarrow a-b i\) are the only automorphisms of \(\mathbb{C}\).