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Questions tagged [galois-extensions]

For questions about Galois extensions of fields. We say that an algebraic extension $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

A Galois extension is a field extension that is normal and separable. See Wikipedia for more information.

738 questions
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Galois group of $x^5-x-1$ over $\Bbb Q$

I am trying to compute the Galois group of $x^5-x-1$ over $ \Bbb Q$. I've shown that this polynomial is irreducible over $\Bbb Q$, by showing that it is irreducible over $\Bbb Z_5$. Let $F$ be the splitting field of $x^5-x-1$ over $\Bbb Q$. This…
blancket
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Inverse Galois problem

The inverse Galois problem conjectures that every finite group is (isomorphic to) the Galois group of some Galois extension of $\mathbb Q$, however it is not known. My question is: what is the smallest finite group such that it is not known…
Régis
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Let $F$ be a Galois extension over $\mathbb{Q}$ with $[F:\mathbb{Q}]=2^n$, then all elements in $F$ are constructible

Let $F\subseteq\mathbb{C}$ be a Galois extension of $\mathbb{Q}$ such that $[F:\mathbb{Q}]=2^n$; then all elements in $F$ are constructible. Added. Here is what I have so far. Since there exist the finite normal extension $F\subseteq\mathbb{C}$…
user7097
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Galois Groups of Finite Extensions of Fixed Fields

I am trying to prove the following proposition: Let $L$ be an algebraically closed field, $g \in Aut(L)$ and $K=\{x \in L \; | \; g(x)=x\}$. Show that every finite extensions $E/K$ is a cyclic Galois extension. So far my thoughts have led me to…
Holdsworth88
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Let $F \supset E \supset K$, $E/K$ and $F/E$ Galois and every $\sigma \in \mathrm{Aut}_K(E)$ extendible to $F$. Then $F/K$ is Galois.

Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. Show that $F$ is Galois over $K$. The definition…
cyc
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Method of showing maximal number of intermediate field extensions of a Galois extension with given degree

The task is the following: Show that a Galois extension $L/K$ of degree $45$ has got at most $12$ intermediate field extensions. Below I present a proof. I seek a more general method for this kind of problems. Using Galois correspondence, each…
B.Swan
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Infinitely many number fields

I wonder if it is known that there are $\textit{infinitely}$ many number fields $F$ (up to isomorphism) with fixed degree $[F:\mathbb{Q}]=n$ and fixed a transitive group $G$ of $S_n$ such that $G=\textrm{Gal}(F^c/\mathbb{Q})$ (if we assume inverse…
Thomas
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Galois group of a function field over finite field

I have a question about the structure of this Galois group that I can't understand: suppose that $p>2$ is prime and $q$ is any power of p, and we have these two function…
Soroush Salehin
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'Simple' Proof: Infinitely Many Galois Fields of Fixed Degree

TLDR; Is there an 'elementary' argument to prove the following: Claim: Given an integer $d>1$, are there infinitely many distinct Galois extensions $K/\mathbb{Q}$ with $[K \colon \mathbb{Q}]= d$? Elementary in the sense that students new to…
mathematics2x2life
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Ways of finding primitive element of separable extension $\Bbb{Q}(\sqrt[4]{2},i)$ over $\Bbb{Q}$.

Consider the field extension $L=\mathbb Q (\sqrt[4] 2 ,i)$ over $\mathbb Q$. This extension is separable as we know over a field of characterstic $0$. Now according to the primitive element theorem there exist an element $\gamma$ such that…
Theorem
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Decrease in algebraic degree on multiplying a real number with a root of unity

Let $\omega$ be a $\textbf{root of unity}$ with algebraic degree(degree of its minimal polynomial over $\mathbb{Q}$) $d_1$ over $\mathbb{Q}$ and $r$ be a $\textbf{real number}$ with algebraic degree $d_2$ over $\mathbb{Q}$. Can $r\omega$ have…
nikhil_vyas
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What is an Archimedean prime?

I need to learn some infinite ramification theory and I am stuck with understanding it. I understand that we consider the order of the inertia group as the ramification index, and if the inertia group is trivial, it is unramified. Mostly I am…
Izzy Garcia
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For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z\left[\sqrt n\right]$ stay in $\mathbb Z\left[\sqrt n\right]$?

This problem has a geometric origin which I'll outline below, but I believe the concepts and explanation are algebraic. Given a function on triples $$K((x,y,z))=x+y+z+2\sqrt{xy+yz+zx}$$ we build a general recursion as follows. Start with an integer…
brainjam
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Showing $K(\sqrt \alpha)/F$ is Galois if and only if $\sigma(\alpha)/\alpha$ is a unit and a square.

I would like help solving is the following problem: Assume that $K/F$ is a finite Galois extension and $\text{char} F \neq 2$. Let $G:= \text{Gal}(K/F)$ be its Galois group and let $\alpha \in K^\times$. Show that $K(\sqrt{\alpha})/F$ is a Galois…
James Leslie
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Quadratic subfield of $\mathbb{Q}(\zeta)$

It is well known that there is only one quadratic subfield of $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $p$th root of unity for some prime $p$. I wonder if the same is true for $\mathbb{Q}(\alpha)$ with $\alpha$ a $n$th primitive root of…
Philomeno
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