Questions tagged [galois-theory]

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 (galois-connections).

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Odd field extensions, possible?

I was under the impression that for all field extensions the degree would be $2^n$ for some $n \in \mathbb{Z}$. Am I mistaken? I am wondering as I was faced with the problem of showing that: for $[F(\...
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Definition of field extensions: Can we talk about extensions like $\mathbb{F}_3(5)/\mathbb{F}_3$?

First of all, sorry for this absolute beginner question. It has been bothering me a lot and I think I am just getting something conceptually wrong here. I am learning about extension fields and use ...
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Proof verification: given a field $k$, any finite group can be represented as $\rm Gal(E/F)$ where $k\subset F\subset E$ and $E/F$ is Galois.

Let $k$ be an arbitrary field and $G$ be an arbitrary finite group. It seems to me that one can construct fields $F$ and $E$ such that $k\subset F\subset E$ and $\rm Gal$$(E/F)$ is isomorphic to $G$ ...
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What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation)

What is the value group of $ \Bbb{F}_q((t^{1/p^n}))$ ?(Detailed caluculation) I'm not good at calculating value group, and my only tactics is to calculate the value of each element. $|t^{p^{1/n}}u|=1/...
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Galois Group of extension with base not of degree 2

Problem: Show that if $K$ is a field such that $charK\neq 2$ and $L/K$ is Galois then $L/K$ is cyclic of degree $2$ if and only if there exists $\theta \in L$ such that $\theta\notin K$ but $\theta^2\...
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The degree of simple radical extensions

Let $m \le n$ be positive integers. Does there necessarily exist a field extension $K/F$ such that $[K:F] = m$ and $K = F(u)$ for some $u \in K$ satisfying $u^{n} \in F$? In other words, given $m \le ...
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What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group?

What is the perfect closure of $ \Bbb{F}_p((t))$ and its value group ? I think perfect closure is $ \Bbb{F}_p((t^{1/p^∞}))$, and I think if so, $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$? $ \Bbb{F}_p((t^{1/p^...
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Any element of K(X) which is not in K is transcendental over K

I was solving this exercise from Hungerford but apparently I couldn't understand it properly since I have a counter example(?). The proposition is this: Now, when I choose $K = \mathbf{Q}$ and $K(x_1,...
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Is the Galois group of $\mathbb{Q} \left(\sqrt{5}+\sqrt{7}+i \right)$ isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times\mathbb{Z}_{2} $?

First of all, I proved that $\mathbb{Q}(\sqrt{5}+\sqrt{7}+i)=\mathbb{Q}(\sqrt{5},\sqrt{7},i)$.Found that $|G|=|Gal(\mathbb{Q}(\sqrt{5},\sqrt{7},i)|=[\mathbb{Q}(\sqrt{5},\sqrt{7},i):\mathbb{Q}]=8$. ...
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Why is $J_{1} \cdots J_{M}$ an integer in Lindemann-Weierstrass theorem?

I was going through the proof of Lindemann-Weierstrass theorem written in Chapter 4, page 16 of the following book: Transcendental Numbers by M. Ram Murty and Purusottam Rath and I can't see how $J_{1}...
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Extend an action of indeterminates to an action of field of rational functions

In this text in the theorem 1.15 the author constructs a Galois extension using group action. In his construction, I can see why $G$ can be seen as an automorphism group of $K$, but I cannot see why ...
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Showing that $x^3 - t$ is irreducible over $\mathbb{F}_3(t)$

I was reading the post Is $\mathbb{F}_3(t,t^{1/3})/\mathbb{F}_3(t)$ a normal extension? Is it separable? I do not understand, why we can use Eisenstein's criterion to show that $x^3 - t$ is ...
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Coinvariants of coaction $a\otimes b \mapsto \sum{\sigma_i(a)}\otimes \sigma_i(b)\otimes \sigma_i^*$

I've been studying Hopf-Galois Theory and currently I'm trying to understand some examples by writing all the explanations step by step by myself. The example I'm interested now is the classical ...
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Bhargava’s proof of van der Waerden conjecture: how to use Hilbert irreducibility to show almost all polynomials have Galois group $S_n$

On the first page of his paper where he proves van der Waerden’s conjecture, Bhargava mentions that Hilbert’s irreducibility theorem shows that the number of monic integer polynomials of degree $n$, ...
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Vanishing sums of integral linear combinations of roots of unity

Let $\{ \xi^{i} \}_{i=1}^{n}$ be $n$-th roots of unity for some positive integer $n$. It is well known that if $n$ is a prime integer, there will be $n-1$ primitive $n$-th roots of unity which are ...
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Why does Galois theory most naturally take place in the context of fields?

At least as far as I can tell, historically Galois theory was a more computational tool than it appears now, and https://hsm.stackexchange.com/questions/8099/how-did-the-modern-understanding-of-galois-...
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A non-cyclic quartic

Given the quartic $x^4 - 2c x^3 + (c^2 - d^2) x^2 + 2a^2 c x - a^2 c^2 = 0$ for integers $a$,$b$ and $c$ where $d^2 = a^2 + b^2$. Looking at the Galois group $G$ of this quartic, we can WLOG assume ...
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How to get a polynomial corresponding to a solvable Galois group of order 60?

Of course, we know the $A_5$ of $60$ order is an unsolvable group. But as the wiki here, there are also $12$ solvable groups in the same $60$ order still: Then I have generated many many irreducible ...
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A lemma about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. (What he calls a 'pure extension' is commonly called 'radical extension' by most authors.) I am confused by ...
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Is there a general way to compute fixed fields by using other (already computed) fixed fields?

I know this question is a little vague, so I try to formulate the problem a bit more precisely. Let $L/K$ be a finite Galois extension and $G$ the corresponding Galois group. We also suppose that $G$ ...
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Lemma A-5.19 about the Galois group of radical extensions in Rotman's book Advanced Modern Algebra

Here is the proof of Lemma A-5.19 in Chapter A-5 of Rotman's book Advanced Modern Algebra. It is about the characterization of the Galois group of pure extensions (which are mostly called radical ...
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find all irreducible polynomials of degree 2 and 3 over Z5

I would like to find all irreducible polynomials of degree 2 and 3 with coefficients in Z5. I know that the polynomial (x^5)^n - x equals with the product of all monic irreducible polynomials of ...
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Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to its action on $\mathbb{C}_p$?

Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/\mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $\mathbb{C}...
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van der Waerden's proof that a monic $p(x) \in \mathbb{Z}[x]$ has Galois group $S_n$ with probability 1

This paper mentions that van der Waerden proved some results on the density of monic integer polynomials with Galois group the symmetric group $S_n$ in 1936. I have found van der Waerden's original ...
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Can one show that in a certain sense, "most" polynomials have Galois group $S_n$? [duplicate]

Intuitively, it seems that given a random irreducible $p(x) \in \mathbb{Q}[x]$ of degree $n$, the Galois group of $p(x)$ over $\mathbb{Q}$ should be $S_n$. Otherwise, if $\alpha_1,...,\alpha_n$ are ...
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Is the following statement about splitting fields true?

Let us take a look at the following definitions: Let $F$ be a field and $f(x)\in F[x]$ then a field extension $E$ of $F$ is said to be the splitting field of $f$ over $F$ if $f$ splits completely in $...
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If $f(x)$ is irreducible, is $f(x^k)$ irreducible?

Let $f(x)\in\mathbb{Z}[x]$ be an irreducible polynomial of degree $\ge 2$. Is it true that $f(x^k)$ is irreducible for $k\ge 2$? If not true, under what hypothesis, we can gurantee positive answer? ...
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Equivalent definitions of Galois extension. [duplicate]

I am studying Galois theory.I encountered some equivalent definitions of Galois extension: The following are equivalent for a finite extension: $1. E/K$ is a splitting field of separable polynomial ...
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Solvability of the quintic by radicals - missing step?

A theorem due to Galois asserts that a polynomial $f\in F[x]$ can be solved by radicals iff. the Galois group of $f$ is a solvable group. In my lecture notes as a corollary I have the following: The ...
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Every finite separable extension is contained in a Galois extension

I am having trouble understanding the following proof: Claim: Let $K/F$ be a finite separable field extension. Then $K$ is contained in a Galois extension $K \supset L \supset F$. Proof: Since the ...
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Let $\alpha = \sqrt{5 + \sqrt{5}}$. Prove that $\mathbb{Q}(\alpha)/\mathbb{Q}$ is Galois and find the Galois group [duplicate]

So far, what I have figured out is that $\alpha$ is a root of the polynomial $f(x) = x^4 - 10x^2 + 20 \in \mathbb{Q}[x]$, and that $f$ has 4 distinct roots: $$ \alpha = \alpha_1 = \sqrt{5 + \sqrt{5}},...
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Compute the Galois group for $f(x) = (x^2-p)(x^2-q)(x^2-pq)$ over $\mathbb{Q}$ and determine all subfields of splitting field

For $f(x) = (x^2-p)(x^2-q)(x^2-pq) \in \mathbb{Q}[x]$ where $p\neq q$ are primes I need to compute the Galois group for $f$ over $\mathbb{Q}$ and determine all subfields of the splitting field. Here ...
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What is the lattice diagram of the Galois extension $\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$?

Consider the Galois extension $K:=\mathbb{Q}(\sqrt 2, \sqrt 3, \sqrt{(2+\sqrt 2)(3+\sqrt 3)})/\mathbb{Q}$. As it is clear, it is of degree $8$ extension. So the Galois group is a group of order $8$. I ...
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Is $i$ an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$?

I am trying to check if $i$ is an element of $\mathbb{Q}(\sqrt[4]{5},w)$ where $w=e^{2πi/3}$. How can I check if this is the case? Would it be correct to express my field as $a\mathbb(\sqrt[4]{5}) + b ...
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Equality of divisors under Galois action in Silverman

I am trying to understand a formula in Silverman's The Arithmetic of Elliptic Curves. Let $K$ be a field, $\overline{K}$ an algebraic closure of $K$, and $C$ be a curve defined over $K$, and $\sigma \...
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5 votes
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How the Frobenius behaves under the change of base field

Let $k'/k$ be an extension of finite fields, $X$ be a scheme over $k'$ and thus over $k$. Then $X\otimes_{k} \bar{k}$ is $n = [k':k]$ disjoint union of $X\otimes_{k'} \bar{k}$. How the Frobenius of ...
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3 votes
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The Galois group of an irreducible quartic whose roots are pairwise rationally independent

Let $f \in \mathbb{Q}[x]$ be an irreducible quartic, $L/\mathbb{Q}$ its splitting field. Label its roots $\alpha_1, \dots , \alpha_4$. Suppose that $$\mathbb{Q}(\alpha_i) \, \cap \, \mathbb{Q}(\...
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Compact-open topology and Krull topology on Galois groups

Let $E/F$ be a Galois extension and consider $E$ as a discrete topological space. Denote $G^{KO} = \operatorname{Gal}(E/F)^{KO}$ the Galois group endowed with the Compact-open topology in $\...
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Proving that the first Galois cohomology group is direct limit of finite quotients

This question comes from Silverman's Arithmetic of Elliptic Curves, specifically the appendix on Galois cohomology. I am a cohomology beginner, interested (for now) in understanding just enough to get ...
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Why the roots of cubic equations can be written in three parts from the perspective of group theory

The root cubic equation $$a x^3+bx^2+cx+d=0$$ can be written as $$x_i = \sqrt[3]{A_i} + \sqrt[3]{B_i}+C,~~i\in\{0,1,2\}$$ And $A_i, B_i$ can be rotated to coincide. How to understand this form from ...
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Is it an extension of Galois?

Let $F \subset E \subset K$ be field extensions and $[E:F]=2$. If K/F is a Galois extension then E/F is Galois. This is true? Does anyone know how to prove it? I really appreciate any help.
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Strategy to prove continuity of group action $G×L→L$

Let $G$ and $L$ be topological groups, and $G$ acts on $L$ via the map $f:G×L→L$. I want to prove $f$ is continuous. From definition,$f$ is continuous if only if for arbitrary open subset $U$ of $L$, $...
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Galois group of $\mathbb{Q}(\sqrt{a+d\sqrt{b}},\sqrt{a-d\sqrt{b}})$.

I was reading these lecture notes of Miles Reid: https://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf on page 47, he writes example 3.21 of $\mathbb{Q}(\sqrt{a+\sqrt{b}},\sqrt{a-\sqrt{b}})$, but he ...
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Does the discriminant of a polynomial depend on the underlying field?

Say I want to show the discriminant of $X^4 + rX + s \in \mathbb{Q}[X]$ is $-27r^4 + 256s^3$. I can do this by showing it's a symmetric polynomial in r,s so by total degree in the roots it's a linear ...
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1 vote
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Why does solving polynomials of degree $n$ via radicals break down at $n=5$, instead of at, say, $n=7$ or $n=3074$?

I know from questions such as this one or this one that Galois theory has proven there exists no "closed formula" (ie. via radicals) for solving quintic equations. My question is more of a ...
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Resolvents of quartic polynomials

Let $f(x)\in\mathbb{Q}[x]$ be monic irreducible of degree $4$, with $$ f(x)=(x-\alpha_1)(x-\alpha_2)(x-\alpha_3)(x-\alpha_4) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (\alpha_i\in\mathbb{C}) $$ In ...
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2 votes
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Degree of subfield fixed by single automorphism

Let $L/K$ be a finite Galois extension. If $\sigma \in \mathrm{Gal}(L/K)$ has order $d$, is it the case that $$L^\sigma := \{ \ell \in L : \sigma(\ell) = \ell\}$$ satisfies $[L^\sigma:K] = d$? This ...
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Practicing degree arguments for field extensions

Right now I'm just trying to get a better grip on my understanding for prelims. The question is as follows: Suppose we have the polynomial $x^6 -5$, Prove this is irreducible over $\mathbb{Q}$ What ...
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Is it possible to produce identically-behaving binary extension fields using different irreducible polynomials?

Let $GF(2^m)$ be a binary extension field with constructing polynomial $f(z)$ be an irreducible, primitive polynomial over $GF(2)$. Is there any possibility that two (or more) different $f(z)$ can ...
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1 vote
1 answer
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Degrees of field extensions

Find the degrees of $\Bbb{Q(a)/Q, Q(a,b)/Q, Q(a,b,c)/Q}$ where $a,b,c$ are the roots of $x^3+x-1$ where a is real. To solve this problem, I noticed that $x^3+x-1$ is irreducible. Then why isn’t the ...
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