Cyclotomic Fields

Let \(\zeta _8 = \exp (2\pi i/8)\), complex eight root of one. Use Galois theory to determine all subfields of \(\mathbb {Q}(\zeta _8)\).

Describe a primitive generator for the degree two extension of \(\mathbb {Q}\) inside the extension \(\mathbb {Q}(\zeta )\), where \(\zeta :=e^{2\pi i/7}\). Justify that the extension you identify is of degree two.

Prove that \(\mathbb {Q}(\sqrt [3]{2})\) is not a subfield of any cyclotomic field over \(\mathbb {Q}\).

Let \(E:=\mathbb {Q}(\zeta +\zeta ^{-1})\) where \(\zeta \in \mathbb {C}\) is a primitive \(n\)th root of 1, for some integer \(n\ge 3\). Determine the Galois group of \(E\) over \(\mathbb {Q}\).

Let \(p\) be a prime integer, and set \(f(x)=x^{p-1}+x^{p-2}+\cdots +x+1\). Suppose a prime integer \(q\) divides \(f(a)\) for some integer \(a\), prove that either \(q=p\) or \(q\equiv 1\bmod p\).

Use this to prove that the arithmetic sequence \(1,1+p,1+2p,\ldots \) contains infinitely many prime integers.

Let \(\alpha \) be a complex root of \(x^6+x^3+1\). Determine all field homomorphisms \(\mathbb {Q}(\alpha )\longrightarrow \mathbb {C}\).

Find all positive integers \(n\) such that \(\cos (2\pi /n)\) is rational.

Let \(\alpha \) be a complex root of \(x^6+3\). Set \(K=\mathbb {Q}(\alpha )\).

1. Prove that \(K\) contains a primitive 6-th root of unity.
2. Compute the Galois group of \(K\) over \(\mathbb {Q}\).

Let \(\zeta \) be a primitive 16-th root of unity over a field \(K\). Determine \([K(\zeta ):K]\) when \(K\) is:

1. \(\mathbb {F}_7\)
2. \(\mathbb {F}_9\)
3. \(\mathbb {F}_{17}\)

Let \(F\) be a field that contains a primitive \(n\)th root of unity. Show that if \(a\) is an element of \(F\) and the field \(E\) is obtained from \(F\) by adjoining an \(n\)th root of \(a\), then \(E\) is a Galois extension of \(F\) with cyclic Galois group.

Let \(L=\mathbb {Q}(e^{2\pi i/8})\). Determine all subgroups of the Galois group \(\operatorname {Gal}(L/\mathbb {Q})\), and the corresponding intermediate fields.

Take a regular \(n\)-sided polygon inscribed in a circle of radius 1. Label the vertices \(P_1,\ldots ,P_n\), and let \(\lambda _k\) be the length of the line joining \(P_k\) and \(P_n\) for \(1\leqslant k\leqslant n-1\). Prove that \[ \lambda _1\cdots \lambda _{n-1}=n. \]

Let \(\omega =e^{2\pi i/3}\). Let \(\sigma \) and \(\tau \) be automorphisms of \(\mathbb {C}(x)\) which fix \(\mathbb {C}\) and satisfy \[ \sigma (x)=\omega x,\qquad \tau (x)=1/x. \] Prove that the group \(\langle \sigma ,\tau \rangle \) is isomorphic to the symmetric group \(S_3\), and determine the subfield of \(\mathbb {C}(x)\) that is fixed by this group.

Let \(\zeta _8=e^{\frac {\pi }{4}i}\). Determine the Galois group of \(K=\mathbb {Q}(\zeta _8)\) and then determine all quadratic extensions of \(\mathbb {Q}\) contained in \(K\).

Let \(k\) be a field containing a primitive \(n^{\mathrm {th}}\) root of unity. Show that for any \(a\in k\) the Galois group of \(X^n-a\) over \(k\) is abelian.

Let \(E\) be a finite extension of the rationals. Show that the number of roots of unity in \(E\) is finite.

Show that if the field \(F\) contains a primitive \(n^{\mathrm {th}}\)-root of unity, then the Galois group of the polynomial \(x^n-a\in F[x]\), \(a\in F\), is abelian.