Questions tagged [cyclotomic-fields]
Cyclotomic fields are fields where a primitive root of unity is added to the rational numbers. These fields are common in algebraic number theory.
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Totally $p$-adic elements in cyclotomic extensions of $\mathbb{Q}$
Let $p$ be a prime number and fix $\bar{\mathbb{Q}}$ an algebraic closure of $\mathbb{Q}$. An element $\alpha \in \bar{\mathbb{Q}}$ is called totally $p$-adic if $p$ splits completely in $\mathbb{Q}(\...
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Isomorphism $\mathbb Z[\omega]/(1-\omega)^2\cong (\mathbb Z/(p))[x]/(1-X)^2$, $\omega$ is the $p-$th root of unity.
Im reading the following proof of Fermat's Last Theorem from Keith Conrad
https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf
On page 5 he mentions that $\mathbb Z[\omega]/(1-\omega)^2\cong (\...
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Galois Group of $\mathbb Q$ Surjects onto Certain Cyclotomic Extension
Let $\ell$ be a prime and let $G_{\mathbb Q, \ell}$ denote the Galois group of $\mathbb Q(\mu_{\ell^\infty})/\mathbb Q$, the extension of $\mathbb Q$ formed by adjoining all primitive $\ell^n$th roots ...
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Linear Dependence of Primitive Roots of Unity
Consider the cyclotomic field $\mathbb{Q}(\zeta_n)$. We know that the set of primitive roots $\Pi_n=\{\zeta_n^m:(m,n)=1\}$ generates $\mathbb{Q}(\zeta_n)$ as a field. However, what happens when we ...
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$\sqrt{p^*}$ contained in $\Bbb Q(\zeta_p)$ [duplicate]
Let $p$ be a prime. I read several times e.g. here that the square root $\sqrt{p^*}$ (where $p^*:=(-1)^{(p-1)/2}p$) is contained in $\Bbb Q(\zeta_p)$. Is there a "standard argument" see it?
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Check when a positive square Root $\sqrt{d}$ is contained in Cyclotomic Field
Let $\zeta_n $ be a primitive root of unity generating the cyclotomic field $\Bbb Q(\zeta_n)$. Is/are there quick and/or "standard" techniques" to check if a given real quadratic roots $...
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Mod $\ell$ Cyclotomic Character Evaluated at Frobenius Away From $\ell$
Let $\ell$ be a prime, let $p \neq \ell$ be prime, and let $P$ be a prime lying above $p$ in $\overline{\mathbb Z}$, the ring of integers of $\overline{\mathbb Q}$. If $\chi_\ell$ is the $\ell$-adic ...
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Why should a splitting field of a polynomial over $\mathbb{F}_p[X]$ be a cyclotomic extension?
I have just been doing a Galois theory exercise and one part of the exercise requires me to explain why, if $f \in \mathbb{F}_p[X]$ is a polynomial of degree $n$, its splitting field $L$ is a ...
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Calculate explicitly $a$ satisfying $(1-\zeta_p)^{p-1}= pa$ inside $\mathbb{Z}[\zeta_p]$
Let $p \neq 2$ be a prime and $\zeta_p$ be a (nontrivial) root of unity. It is well known that $(1-\zeta_p)^{p-1}$ is divisible by $p$ inside $\mathbb{Z}[\zeta_p]$, i.e., there exists $a \in \mathbb{Z}...
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A Formula for the Norm in the Cyclotomic Field of Degree 5
So you all know and love, the Gaussian integers have a rather neat-looking norm: given $a+bi$, then $N(a+bi)=a^2 + b^2$. When it comes to the cyclotomic integers of degree 3, i.e. the domain $\mathbb{...
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The primitive $n^{th}$ roots of unity form basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity iff $n$ is square free
Prove that the primitive $n^{th}$ roots of unity form a basis over $\mathbb{Q}$ for the cyclotomic field of $n^{th}$ roots of unity if and only if $n$ is square free
I think I have the $(\Rightarrow)$...
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How to calculate Gal$(F(\mu_{p^\infty})/F(\mu_p))$ for a number field $F$?
Let $F$ be a number field. Recall that we define $$F(\mu_{p^\infty})=\bigcup_{n=1}^{\infty}F(\mu_{p^n}).$$
I want to calculate the group Gal$(F(\mu_{p^\infty})/F(\mu_p))$. I know that this is supposed ...
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When is $\sqrt n$ in $\Bbb Q[\omega_m]$?
Given a positive integer $n$ and a primitive $m^{th}$ root of unity $\omega_m$ over $\Bbb Q$, how could one determine if $\sqrt{n}$ lies in $\Bbb Q[\omega_m]$?
In the case of $n=p>0$ being an odd ...
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If $\sqrt{n}$ is contained in $\Bbb Q[\omega_m]$, so does $\sqrt{p}$ for any $p|n$.
I know that every square root $\sqrt{n}$ of an integer is contained in some cyclotomic extension $\Bbb Q[\omega_m]$. If we know $\sqrt{n}\in\Bbb Q[\omega_m]$ and that $n$ is square-free, can we ...
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Why is $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ iff $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$?
I am reading David Zywina's "Elliptic curves with maximal Galois action". For a number field $K$, he defines $\mathbb{Q}^{\text{cyc}} \subset \overline{K}$ to be "the" cyclotomic ...
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