# 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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### Why p-cyclic extension iff $p^m$-cyclic extension $\forall m$

A theorem is stated as follows. For a field $F$ of characteristic $p$, $F$ has a $p$-cyclic extension if and only if for every positive integer $m$, $F$ has a $p^m$-cyclic extension. I wonder if ...
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### Show that this Galois group is solvable (Base case)

So I've been reading a proof of Theorem : If E/K is a normal and radical field extension then Gal(E/K) is a solvable group. The author then goes on a proof by induction but he shrugs off the base ...
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### Degree of the Kummer extension - computing

I have a question. I want to calculate the degree of the Kummer extension. $D_{g}(k) = (\mathbb Q(g^{1/k}, e^{2\pi i /k}) : \mathbb Q)$ [Page 3 in this publication] What is the easiest way to do ...
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### $E/F$ is an extension with radicals of order $n$

We have that $E/F$ is an extension Kummer of degree $n$ and that $F$ contains a $n$-th unit root $\omega$ with $\text{ord} (\omega)=n$. I want to show that $E/F$ is an extension with radicals of ...
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### minimal polynomial in Kummer extension

Let $n>1$ be an integer. Let $K$ be a field such that $n$ does not divide the characteristic of $K$ and $K$ contains the $n$-th roots of unity. Let $\mu_n\subseteq K$ be the set of $n$-th roots of ...
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### Real solution of Kummer's equation

I'm trying to solve a particular differential equation: $$(c*z+d)*y''(z)+c*y'(z)=(a*z+b)*y(z)$$ wherea, b, c and d are constant and y is a function of z. This is a particular Sturm-Liouville ...
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### Irreducible polynomial of degree $5$ in $L[X]$ using Kummer's theory

Let $L=\mathbb{F_2}(\theta)$ where $\theta$ is a root of $X^4+X+1$ I am trying to solve the following consecutive questions in Galois Theory: Prove that $L$ has degree $4$ over $\mathbb{F_2}$ and ...
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### On a certain representation of the Galois group of $X^n-a$ from Lang's Algebra [duplicate]

I am having trouble understand theorem $9.4$ of Chapter $6$ of Lang's Algebra (pg. 300-301). The setup is a we have a field $k$ of characteristic not dividing $n$. We know that the splitting field ...
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### “Any radically closed field contains all roots of unity”

I've seen the statement "any radically closed field contains all roots of unity." Though the term "radically closed field" doesn't seem to be extremely common, I'm fairly confident that it means that ...
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### Diophantine equation resembling FLT

I was wondering if the equation $x^p+y^p=2z^p$ has been studied. For small cases it is seen that the only solutions are trivial: $x=y=z$. There are probably methods to solve this for regular ...
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I am trying to understand Kummer theory and I wish to apply it to global fields, so our field $K$ containing $\mu_m$ should be $\mathbb{Q}(\zeta_m)$. Let $B$ be a subgroup of $K^{*}$ containing $K^{*... 1answer 87 views ### image of Kummer isomorphism Let$K$be a finite extension of the field of$p$-adic numbers which contains the$p$-roots of unity. There is an isomorphism$K^{\times} / (K^{\times})^p \to \mathrm{Hom}(\mathrm{Gal} (\bar{K} / K), ...
I was reading Serre's paper "Sur les Représentations Modulaires de Degré $2$ de Gal($\bar{\mathbb{Q}}/\mathbb{Q}$)" where he states his modularity conjecture (which is now a theorem). Following his ...
### Galois extension with Galois group $A_4$
Suppose $\operatorname{char} F \neq 2$ and $K/F$ is a degree three Galois extension with $\operatorname{Gal}(K/F)\cong \mathbb{Z}/(3)$. Is there a bijection between extensions $N/F$ with Galois group \$...