# Tagged Questions

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### 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 $n$. ...
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### 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, ...
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### 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 $\phi \in$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...