1
vote
1answer
57 views

Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
4
votes
1answer
312 views

A finite field extension $K\supset\mathbb Q$ contains finitely many roots of unity

I must show that any finite field extension $K\supset \mathbb Q$ can only contain finitely many roots of unity. I reasoned in the following manner: Let $n<\infty$ be the degree of $K$ over ...
0
votes
1answer
100 views

How can one show that no cyclotomic field contains $\sqrt[3]{2}$?

This is a paraphrase of a problem from an old exam that I'm going over. Show that for all positive integers $t$, when $\omega$ is a primitive $t$-th root of unity, $\sqrt[3]{2}$ does not lie in ...
1
vote
1answer
486 views

Determine splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$

Determine the splitting field $K$ over $\mathbb{Q}$ of the polynomial $x^3 - 2$ Also determine the basis over $\mathbb{Q}$ and its degree. Can I do this using only first principles?
4
votes
3answers
851 views

Quadratic subfield of cyclotomic field

Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
0
votes
1answer
225 views

A radical extension with a non-radical subextension

For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely ...
5
votes
2answers
306 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
0
votes
2answers
58 views

The sum of the first and last $k$-th roots of unity

Let $\zeta_k$ be the first complex primitive $k$-th root of unity, i.e. $\zeta_k = e^{2\pi i/k}$. Then the last complex primitive $k$-th root of unity is given by $\zeta_k^{k-1} = \zeta_k^{-1} = ...
2
votes
1answer
129 views

Fields modulo $n$-th powers, discrete valuations and roots of unity

I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with ...