Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
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439 views

The radical solution of a solvable 17th degree equation

(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients: $$\begin{align*} x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
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59 views

Field-theoretic description of fixed field of central subgroups?

Given a Galois extension $E/F$ with Galois group $G$, and a subextension $E/K$ with Galois group $H$, is there a "field-theoretic" characterization of when $H$ is central (i.e. $H\leq Z(G)$)? By ...
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205 views

On solvable quintics and septics

Here is a nice sufficient (but not necessary) condition on whether a quintic is solvable in radicals or not. Given, $x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ If there is an ordering of its roots such ...
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126 views

Relationship between intersection and compositum of fields

This issue came up in a number theory lecture today. Let $K$ be a number field and let $L/K$ be an abelian (finite Galois) extension. Then there exists a primitive $m$th root of unity $\zeta_m$ so ...
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78 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
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284 views

Proof that a finite separable extension has only finite many intermediate fields

Let $E/F$ be finite separable extension. Is there any proof of the fact that there are only finitely many intermediate fields without using primitive element theorem or fundamental theorem of Galois ...
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464 views

“The Galois group of $\pi$ is $\mathbb{Z}$”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question: The Galois group of $\pi$ is $\mathbb{Z}$. In what sense/framework is ...
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69 views

Generalization of Kummer isomorphism?

Let $p$ be a prime number and denote by $\mathbb{F}_p(1)$ the one dimensional vector space over $\mathbb{F}_p$ endowed with an action of $G:=Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ via the mod $p$ ...
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200 views

Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
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42 views

Is there any relation between Galois solvability and integrability of hamiltonian systems?

Galois theory provides a method and formalism to study solutions of polynomial equations and solvability. Dynamical Hamiltonian systems have a somewhat similar concept of integrability. Since many ...
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75 views

Complex Conjugation as an $F$-Automorphism of $K$?

I am struggling with the following problem: Let $f \in F[x]$ be an irreducible quintic polynomial with splitting field $K$, where $\mathbb{Q} \subseteq F$. Supposing that $f$ has three real roots ...
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98 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
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88 views

Understanding a Proof in Galois Theory Notes

My course lecture notes for Galois Theory make the following (standard) claim about the uniqueness of splitting fields for a given polynomial. I am struggling to understand the proof the lecturer ...
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117 views

Galois Group of $x^5+ 5x^3 + 5x + 1$.

I've been asked to determine the Galois Group of $x^5+ 5x^3 + 5x + 1$. This is what I know so far. 1) The polynomial is irreducible. 2) Its discriminant is $78125=5^7$ Since the discriminant is ...
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194 views

Deriving the Bring-Jerrard quintic using a $cubic$ Tschirnhausen transformation

One can simultaneously eliminate three terms from the general quintic using a quartic Tschirnhausen transformation. The 4th parameter of the quartic allows it to be done in radicals and details are in ...
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157 views

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$

Prove that every algebraic extensions over $\mathbb{R}$ are isomorphic to $\mathbb{R}/\mathbb{R}$ or $\mathbb{C}/\mathbb{R}$ Corollary $3.20$ page $267$ of Hungerford - Algebra: "Every proper ...
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54 views

When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let ...
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87 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
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513 views

Galois group acts transitively

The question I am dealing with is: Let F be a field, $f(x)\in F[x]$ be irreducible and let $N/F$ be normal field extension. Let $$f(x) = g_1(x) \cdot \dots \cdot g_r (x)$$ be the factorization of ...
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Showing that $[\mathbb C(z):\mathbb C(f_0)]=$topological degree of $f_0\in\mathbb C(z)$ as a function from the Riemann sphere to itself

McKean & Moll offer the following sketch of a proof. Let $P(x)=a_0(x)-f_0b_0(x)$ in $\mathbb C(f_0)[x]$ where $f_0(z)=\dfrac{a_0(z)}{b_0(z)}$ is in $\mathbb C(z)$. Then evidently, $\deg P=\#$ ...
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26 views

Properties of the extension $\mathbb Q(\sqrt{a\sqrt{D}})$

Let $a$ be a nonzero rational number, and let $D$ be a squarefree integer not equal to $1$. I want to show that, firstly, the extension $F=\mathbb Q(\sqrt{a\sqrt{D}})$ is not cyclic of degree $4$, and ...
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69 views

Systematically describing the Galois Group and Intermediate Fields

In an exercise in the textbook you are asked to describe the Galois Group and the intermediate fields of the extension $$ L=\newcommand{\Q}{\mathbb Q}\Q(\sqrt 2,\sqrt 3)\supset\Q $$ I have noted that ...
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50 views

Automorphism group of an L-function

I define the notion of Galois class of L-functions as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
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65 views

The Structure of the Galois Group of $K/F$ inside $GL_n(F)$

This question is in two parts. In the first, I give my thoughts and ask for verification that they are correct or whether there are closely related ideas that would improve this intuitive picture. In ...
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102 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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82 views

Relationship between automorphisms of finite etale covers and function fields

Let $X$ and $Y$ be varieties over an algebraically closed field $K$, $\phi:Y \longrightarrow X$ be a finite etale cover , and let $K(X),K(Y)$ be the function fields of $X$ and $Y$ respectively. Then ...
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35 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
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30 views

Determining if a given equation is solvable given a set of ultra-radicals

So suppose someone is armed with the tools of standard arithmetic, exponents (and of course that comes along with roots) AS WELL AS a set of inverses for some polynomials which are not solvable using ...
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64 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
3
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131 views

If $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable.

In all the books that I've checked the use of roots of unity (as hypothesis) is very crucial to prove that if $f(x)\in F[x]$ is solvable by radicals then its Galois group $Gal(E/F)$ is solvable, but ...
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139 views

Identifying the Galois Group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$

I am trying to determine the Galois group $G(\mathbb{Q}[\sqrt{2}, \sqrt[3]{2}]/\mathbb{Q})$. I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just ...
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51 views

Determine $\text{Gal}(\mathbb{Q}(\alpha) / \mathbb{Q})$ where $\alpha = \omega + \omega^7 + \omega^{11}$

Suppose $\alpha = \omega + \omega^7 + \omega^{11}$, where $\omega$ is a primitive root of unity of order 19. Determine the Galois group of $\mathbb{Q}(\alpha) / \mathbb{Q}$. With which other well ...
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47 views

Inverse Galois theory and Hilbert class field

I am not sure if the following questions have an answer. (Question 1) Let $G$ be a finite Abelian group. Is it possible to find an unramified Abelian extension $L/K$ such that $$G \cong ...
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43 views

Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
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48 views

Number field theoretical analogue of Riemann's Existence Theorem

This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1). Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup ...
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61 views

Does cyclic field imply Galois?

I am thinking about the following statement, and I wonder if this is true: Every cyclic field is Galois. (we are in characteristics $0$). I have started with a cubic case and tried to make use of the ...
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Question about why $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$.

I'm trying to follow a solution I'm reading. The idea is to prove $x^{p^n}-x+1$ is irreducible over $\mathbb{F}_p$ when $n=1$ or $n=p=2$. The solution is as follows: If $x^{p^n}-x+1$ is ...
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41 views

Generalized fundamental theorem of Galois theory

In the generalized fundamental theorem of Galois theory, theorem says there is a one-to-one correspondence between the set of all intermediate fields of extension and the set of all $\textbf{closed}$ ...
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158 views

Ramanujan-type trigonometric identities with cube roots, generalizing $\sqrt[3]{\cos(2\pi/7)}+\sqrt[3]{\cos(4\pi/7)}+\sqrt[3]{\cos(8\pi/7)}$

Ramanujan found the following trigonometric identity \begin{equation} \sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+ \sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}= ...
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Bachelor Thesis - Galois Theory Research Topics?

I'm on the last semester of my bachelor's degree (undergrad degree) and I will be writing my thesis next semester. I have talked to a professor at my university and one of the topics he suggested was ...
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104 views

Galois false solution to the quintic equation

I am looking for the false solution Galois gave to the quintic equation before discovering group theory.
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71 views

Parametric 12-deg and 14-deg equations with group $PGL(2,11)$ and $PGL(2,13)$?

We have, $$x^{12} - a x^{11} - 33x^8 + 22a x^7 - 11a^2 x^6 + 363x^4 - 121a x^3 + 121a^2x^2 - 23a^3x - 11^3 + a^4=0$$ $$x^{12} - a x^{11} - 11a x^9 - 44a x^7 - 88a x^5 - 88a x^3 - 32a x - a^2=0$$ ...
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Question about Wantzel's proof of the necessary condition for compass/straightedge constructibility

I'm trying to understand Wantzel's original proof of the necessary condition for constructibility with a straightedge and compass. It's expressed in terms of polynomials rather than field extensions. ...
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137 views

Link between representation theory and Galois theory: Trivial representation in field towers.

Let $K|F$ be a finite cyclic Galois extension of number fields of degree prime to $p$ with Galois group $H$, where $p$ denotes a rational prime. Let $L|K$ denote a pro-$p$-extension (possibly ...
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88 views

How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
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81 views

Splitting field of polynomial over $\mathbb{F}_3$

Let $K$ be the splitting field of the polynomial $(x^3 + x - 1)(x^4 + x - 1)$ over $\mathbb{F}_3$. How many elements does $K$ contain? What I've already done is factoring $(x^3 + x - 1)(x^4 + x - 1)$ ...
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93 views

Gaussian periods

Let p be an odd prime number and $p-1=m d$ a decomposition into positive factors. Then there is a unique cyclic extension $K_d/\mathbb Q$ of degree d ramified only at p. It is contained in the ...
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118 views

Galois group of $x^{4}+ax^{2}+b$

Let $x^{4}+ax^{2}+b\in K[x]$ (with $\mathrm{char}K\neq 2$) be irreducible with Galois group $G$. I have shown that (a) if $b$ is a square in $K$, then $G$ is the Klein 4-group, (b) if $b$ is not a ...
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114 views

Question about solvability of the minimal polynomial of the element in extended field.

Let $F$ have characteristic $0$, and assume we have fields $L\supset K\supset F$. Now suppose $\alpha\in L$ be solvable by radicals over $K$, and the coefficients of the minimal polynomial of ...