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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prove that $[F(\alpha):F]=p$ where $\alpha$ is a root of $x^p-a$ [duplicate]

This is the problem 1.1 from the book, A Gentle Course in Local Class Field Theory. Let $p$ be a prime, let $F$ be a field of characteristic $\neq p$, and let $a \in F^{\times} \backslash F^{\times p}...
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Galois group of $x^p - a$, $a$ - squarefree

Let $p$ be a prime and $a > 1$ be a squarefree positive integer. We wish to understand (at least to some extent) the Galois correspondence in the Galois group $G$ of $x^p - a$. The splitting field ...
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What is $\mathbb{Q}_3(\sqrt{-6})^{\times}/\left(\mathbb{Q}_3(\sqrt{-6})^{\times}\right)^3$?

I'm doing some Galois cohomology stuff (specifically, trying to calculate $H^1(\mathbb{Q}_3,E[\varphi])$, where $\varphi:E\to E'$ is an isogeny of elliptic curves), and it involves calculating $\...
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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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Kummer Generators for Cyclic Extensions

Let $F$ be a field of characteristic not dividing $n$ containing the $n^{th}$ rooth of unity and let $K$ be a cyclic extension of degree $d$ dividing $n$. Then $K = F(\sqrt[n]{a})$ for some nonzero $a ...
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Is two subsequent Kummer extensions again a Kummer one?

Let $F$ be a field of characteristics $p$ such that $\mu_n$, $\mu_m$, $\mu_{nm} \subset F^*$ for $p \nmid n, m \in \mathbb{N}$. Consider two subsequent Kummer extensions $F \subset K \subset L$, that ...
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Cyclic Field Extension of Local Field

Let $K$ be a local field (therefore complete, discrete non-archimedian valuation field) with perfect residual field $\kappa_K:= \mathcal{O}_K/\pi_K$. Assume that $L/K$ is a field extension of $K$ of ...
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Kummer Theory Square Extension

I have a question about an argument used in the proof of Kummer theorem in Kedlaya's notes (see page 8) www.math.mcgill.ca/darmon/courses/cft/refs/kedlaya.pdf Let $K$ be a field which contains a n-th ...
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Kummer Theory Proof

I have a question about a step in proof of Kummer theorem in Kedlaya's notes (see page 8) www.math.mcgill.ca/darmon/courses/cft/refs/kedlaya.pdf We start with a $\alpha \in K* = K \backslash \{0\}$ ...
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Special case of normal basis theorem - Artin 16.M.13

The very last problem in Artin's Algebra, second edition, reads: Let $K/F$ be a Galois extension with Galois group $G$. If we think of $K$ as an $F$-vector space, we obtain a representation of $G$ ...
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Galois extension of exponent $mp^r$ in characteristic $p$

Kummer theory treats Galois extensions of exponents that are not divisible by the characteristic. Artin-Schreier and Witt extend this theory for Galois extensions of exponents $p^r$ in characteristic $...
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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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Kummer-type transformation for a special 2F2 hypergeometric function

I am trying to calculate the multiplication $e^{x} \, _2 F_2(a+1,a+1;a+2,a+2;-x)$, where $a>0$, and $x \in (a-\sqrt{a},a+\sqrt{a})$ approximately. But this expression is not calculable for large $a ...
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kummer extension: converse part

Let $F$ be a field containing a $n$-th primitive root of unity, now given a finite subgroup $G$ of $F^*/(F^*)^n$, let $K$ be the field containing $F$ and $\alpha$ such that $\alpha^n (F^*)^n\in G$. ...
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Deducing Galois group of $t^6 + 3$ over the rationals [duplicate]

I am asked to deduce, in the following order, 3 facts about $G = Gal(f(t) = t^6+3)$ over the rationals: $G$ is order 6 The elements have orders $1,2,3$ $G \cong S_3$ I have figured these out but in ...
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Find the bijection between Kummer's field and Galois subgroup

Let $\mathbb{k}$ is field and $\exists \xi \in \mathbb{k}^*,\ O(\xi) = n$ I need find the bijection between k-isomorphic finite separable normal extension $\mathbb{k}$ (i.e. Kummer's field with an ...
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Is the field generated by adding several p-th roots a Kummer extension

Let $F$ be a field with $(char(F),p)=1$ or $char(F)=0$ containing the roots of unity of order $p$. Consider the following extension of $F$: $$F[a_1 ^{1/p},...,a_k ^{1/p}]$$ and assume $a_i\notin(F^*)^...
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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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Finding a $\gamma$ to define a Kummer extension like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

Previous theory: All the cyclic extensions of order $5$ are $\mathbb{Q}(\zeta_5)(\sqrt[5]{\gamma})/\mathbb{Q}(\zeta_5)$ where $\zeta_5$ is the generator of the group $\left(\mathbb{Z}/5\mathbb{Z}\...
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kummer extension on SAGE

I want to calculate the relative discriminant of field extensions of this kind: $$\mathbb{Q}(\zeta_5)(\sqrt[5]{a})$$ Where $a \in \mathbb{Q}(\zeta_5)$. So I use SAGE and make this calculations: ...
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Neukirch Abstract Kummer Theory. Understanding a Proof.

This question is a sort of follow up to this question, where I introduced context. Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT) The thing I don't understand know is ...
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Neukirch mysterious homomorphism in Abstract Kummer Theory (in his book ANT)

Someone familiar with Neukirch's terminology can understand this post better. Unfortunately it is so much terminology to just explain it here. My question is about what is marked in the picture: Why ...
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Find isomorphism between Kummer's field and field of n-th roots

Let $\mathbb{k}$ be the field of all the $n$th roots of $1$ and $\mathbb{F=k}(\alpha_1, ..., \alpha_n)$ is a Kummer's field over $\mathbb{k}$, where $\alpha^n_i=a_i \in \mathbb{k^*}$, $i=1,\dots,m$. ...
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Galois group for Kummer extension over Cyclotomic extension of $p$-adic field

I am trying to recover the Galois group of the extension $E/F$, where $E$ and $F$ are the fields defined below. $F$ is a finite extension of $\mathbb{Q}_p$, containing a primitive root of unity $\...
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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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Kummer extensions, but then a bit more

I'm struggling with the following past exam question, parts (b) and (c): So, starting with (b): since $E/F$ is Galois of degree $p$, we know that $$\Gamma(E/F)\cong C_p\cong \langle\sigma\rangle,$...
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263 views

Radical extension with root of cubic polynomial

If I take $f(x)$ is an irreducible cubic over $\mathbb{Q}$ with a root $\alpha$ in a splitting field and given that $\mathbb{Q}(\alpha)$ is a radical extension is it true that $\mathbb{Q}(\alpha) = \...
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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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Kummer Theory - Example of Subgroup of $K^{*}$ containing $K^{*m}$ for global fields.

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^{*...
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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), ...
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Serre's Modularity Conjecture — Weight

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 ...
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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 $...