Questions tagged [kummer-theory]
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field.
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Galois degree $p$ extensions of $\mathbb{Q}$ that are ramified only at an odd prime $p$
This question is motivated by a question on Constructive Kroneker-Weber asked by user quanta and an answer by user David E Speyer to this question (which limits itself to regular primes). In order to ...
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Irreducibility of Kummer polynomial over Qp
Let $p$ be a fixed prime and $a \in \mathbb Z_p$.
I am trying to find conditions on $p$ such that the polynomial $f(X):=X^p-a$ is irreducible over $\mathbb Q_p$. I know that if $f$ is reducible, then $...
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Isomorphism between Galois cohomology with coefficients in $\mu_{p^\infty}$ and a Galois group when the base field does not contain roots of unity
Let $p>2$ be a rational prime, $K$ be a CM number field not necessarily containing $\mu_p$, $K^+$ its totally real subfield, $K_\infty/K$ the cyclotomic $\mathbb{Z}_p$ extension, and $K_\Sigma$ the ...
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Cyclotomic Extensions of a Finite Extension of $\mathbb{Q}$
It is well known that if $\zeta_n$ is a primitive root of unity then the extension $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is Galois with Galois Group $(\mathbb{Z}/n\mathbb{Z})^*$. However, if $K$ is an ...
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decomposition in kummer extension
On page 119 of his "Reciprocity Laws" Lemmermeyer gives a theorem about decomposition in Kummer extension. For instance
If $ K=k[\sqrt[\ell]\mu]$ with $\ell$ prime then
if $\ell\not|v_{\frak ...
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Calculate how many fields $L$ such that $\mathbb{Q} \subseteq L \subseteq K$ exist if $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{5})$.
Calculate how many fields $L$ such that $\mathbb{Q} \subseteq L \subseteq K$ exist if $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{3}, \sqrt[3]{5})$.
It is straightforward to note that we only need to find ...
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Kummer extension corresponding to a cohomology class
Let $F$ be a (CM) number field, $k$ be a finite extension of $\mathbb F_p$, $F_S$ be the maximal algebraic extension of $F$ unramified outside $S$. In the 10 author paper p.151, the following are ...
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Kummer Theory: Prove that element with norm = -1 implies a cyclic extension of degree 4
I am reading Pierre Guillot's (excellent) book, A Gentle Course In Local Class Field Theory (2018). The first chapter is on Kummer theory, and I need help with Problem 1.2, part 2 on page 21. This ...
user390351
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Kummer Theory: Prove that element with norm = -1 exists in a certain quadratic number field
I am reading Pierre Guillot's (excellent) book, A Gentle Course In Local Class Field Theory (2018). The first chapter is on Kummer theory, and I need help with Problem 1.2 (part 1) on page 21. This ...
user390351
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Ramification Lemma in a Kummer extension.
I have encountered the following Lemma in a paper:
Lemma:
Let $\ell$ be a prime number and $F$ a field that contains a primitive $\ell$-th root of unity. Let $a\in\mathcal{O}_F$ and $K = F(\sqrt[\ell]...
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An exercise about abelian Kummer extensions
I'm trying to do this problem about abelian Kummer extensions:
Image transcript and my attempts are below:
Let $K/F$ be a Galois extension with Galois group $G=\operatorname{Gal}(K/F)$ of order $n$. ...
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Discriminant of the Kummer extension of p-adics
Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power.
Define by $K:=\mathbb{Q}_p(u^{1/p})$. What is ...
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A subgroup of the Kummer group.
Let $F$ be a field containing all $n^{th}$ roots of unity. An $n$-Kummer Extension over $F$ is a Galois extension of the form $E=F[r_1,...,r_m]$ where $r_i^n\in F$ for all $i=1,...,m$.
From Kummer ...
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Prove that $\left[\Bbb Q\left[\sqrt[d_1]{a_1},...,\sqrt[d_n]{a_n}\right]:\Bbb Q\right]=\prod_{i=1}^nd_i$.
I am trying to prove the following quoted statement.
Theorem 1. Let $r_1,...,r_n$ be real algebraic integers over $\Bbb Q$ with minimal polynomials $x^{d_1}-a_1,...,x^{d_n}-a_n$, respectively. Then ...
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If $L/K$ is abelian with exponent $n$, then $L=K(\sqrt[n]{\Delta})$ with $\Delta=K^*\cap L^{*n}$
I'm trying to understand the proof of this fact, that proceeds in the following way:
$K(\sqrt[n]{\Delta})\subseteq L$ because $\sqrt[n]{\Delta}\subseteq L^*$. For the converse, let's view $L$ as the ...
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