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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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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501 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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157 views

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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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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What is the overall idea of Galois theory?

I am a third year undergraduate, doing a course on Field and Galois theory. Now, while I seem to understand most of the concepts locally, I do not seem to get the 'Whole picture' of what is happening ...
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On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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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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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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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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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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“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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How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of ...
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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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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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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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218 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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Galois group solvable but $f$ not solvable.

I know from a theorem that: Let $F$ be a field of characteristic $0$ and $f(x)\in F[x]$. Then $f(x)$ is solvable by radicals if and only if the Galois group of $f(x)$ is solvable. But what if the ...
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135 views

Any abstract algebra book with programming (homework) assignment?

All: I had studied abstract algebra long time ago. Now, I would like to review some material, particularly about Galois theory (and its application). Can anyone recommend an abstract algebra book ...
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106 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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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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103 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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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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264 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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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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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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615 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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33 views

Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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Splitting field of $1 + x^3 + x^6 + x^9$

I'm studying for a qualifying exam and wanted to know if my reasoning on this problem was correct: Let $L$ be the splitting field of $f(x) = 1 + x^3 + x^6 + x^9$ over $\mathbb{Q}$. Find the Galois ...
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166 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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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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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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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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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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120 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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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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A question concerning cyclic field extensions.

In the study of cyclic extensions we have the following theorem: Theorem Let $K$ be a field containing an $n$-th primitive root of unity $\zeta$. Then the following claims hold: If ...
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Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over ...
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Compute Galois group of these extensions.

I think that I have a problem in the redaction. I have to compute 1) $G=\text{Gal}(\mathbb Q(\sqrt 3,\sqrt 2)/\mathbb Q)$ I know that $\mathbb Q(\sqrt 3,\sqrt 2)$ is the splitting field of ...
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46 views

Solvability of Higher Degree Polynomials by Bring Radicals

A Bring radical of $a$ is any root of the polynomial $x^5+x+a$. It is known that we can solve the quintic if we're allowed to use the Bring radical. Now, I was wondering what happens if we ...
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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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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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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 $W_{L/K}X$ ...
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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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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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48 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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64 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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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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203 views

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 ...