1
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3answers
27 views

Quadratic number fields containing primitive roots of unity

A problem from Michael Artin's Algebra (Second Edition) from Fields: Determine the quadratic number fields $\mathbb{Q}[\sqrt{d}]$ that contain a primitive $n$th root of unity, for some integer $n$. ...
3
votes
2answers
154 views

Discriminant of $x^n-1$

Question is to find discriminant of polynomial $x^n-1$ I consider $f(x)=x^n-1=(x-a_1)(x-a_2)(x-a_3)\cdots(x-a_n)$ Now, ...
1
vote
1answer
53 views

Proof Check: automorphism sends primitive root to primitive root

I was just wondering if this is a valid proof. I am assuming knowledge that if $\phi$ is an automorphism of a numeric field the $\phi$ fixes $\mathbb{Q}$. Also, if $\phi \in$ ...
3
votes
3answers
82 views

Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
1
vote
1answer
59 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 $ ...
1
vote
1answer
57 views

Product of Differences of nth Roots of Unity

I'm trying to show that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=n$$ but am finding it surprisingly difficult. I know by symmetry that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ...
3
votes
3answers
97 views

Roots of unity modulo $p$

Let $f(X)$ be the minimal polynomial of something like $\zeta + \frac{1}{\zeta}$, where $\zeta$ is a primitive $p$-th root of unity for some prime $p > 2$. I'd like to show that $f(X) \equiv ...
1
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0answers
65 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
4
votes
1answer
322 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 ...
2
votes
1answer
62 views

How do you calculate the kernel of the substitution homorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}[e^{i2\pi /r}]$?

I can do that for "easier" substitutions like $x \mapsto i$, but I don't know enough about primitive $rth$ roots yet. Does this require pieces of Galois theory? $r$ is a positive integer. I think ...
3
votes
1answer
99 views

Looking at the intermediate fields of $\Bbb{Q}_7 = \Bbb{Q}(\omega)/\Bbb{Q}$ where $\omega = e^{i2\pi/7}$ .

From Basic Abstract Algebra (Robert Ash): The question that I'm concerned with is number 3, but I will write problems 1 and 2 as well, since they are all related... We now do a detailed analysis ...
4
votes
3answers
872 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 ...
5
votes
2answers
310 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 ...
10
votes
1answer
293 views

Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
10
votes
1answer
180 views

Roots of unity in non-Abelian groups: when do they form subgroups?

I haven't studied group theory in earnest beyond first courses, so my notation may be nonstandard and my question may be a 'standard fact', so bear with me: Consider a group $G$, and for each natural ...