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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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How to show that the field $\mathbb{F}_2(t^{\frac{1}{3}})$ does not contain a third root of unity

I want to find a non-normal extension in characteristic p. The following extension is not normal : $$\mathbb{F}_2(t)\subset \mathbb{F}_2(t^{\frac{1}{3}})$$ It's minimal polynomial is : $X^3-t$ and it'...
muhammed gunes's user avatar
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How can we subtract elements of the unit group so that the difference is still in the unit group

I am trying to find the rank of a matrix formed by the characters of a finite ring over the rationals. The order of the matrix is the number of elements in the unit group by the number of elements in ...
Kofi Amponsah Kwabi's user avatar
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Simplest unsolvable quintic with one real root

I am aware that $t^5-t-1$ is unsolvable, but the proof I have seen involves a theorem linking its Galois group with the Galois group of its reduction mod $p$. If I wish to have a simpler proof (that ...
user21820's user avatar
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Yes/No .An extension having Galois group of order $1$ is normal [closed]

Is the following statement true/false ? An extension having Galois group of order $1$ is normal. I think this statement is false take $K= \mathbb{Q}(\sqrt[5]{3}):\mathbb{Q}$ Galois$(K)=\{e\}$ ...
jasmine's user avatar
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Confusion on isomorphism profinite completion integers

Question: Prove that $\hat{\mathbb{Z}} \cong \prod_{p} \mathbb{Z}_{p}$ is an isomorphism of topological rings. Own attempts: Although this specific question has been asked quite a few times here, ...
ByteBlitzer's user avatar
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How to find the Galois group of an infinite Galois extension? For example $\mathbb{Q}(\mathbb{\sqrt{\mathbb{Q}}})$.

I know that this is a very broad question, but I am especially interest in the case of the Galois extension $\mathbb{Q}(\sqrt{\mathbb{Q}}) = \mathbb{Q}( \{ \sqrt{-1} \} \cup \sqrt{p} \mid p \text{ ...
Cosima's user avatar
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The condition of compactness of subgroup of infinite Galois group

Let $E$ be Galois over $F$, with Galois group $G$. $H$ is a subgroup of $G$. Prove that $H$ is dense in $G$ if and only if for any subfield $K$ of $E$ containing $F$ and $K$ finite and Galois over $F$,...
shwsq's user avatar
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An exercise about abelian Kummer extensions

I'm trying to do this problem about abelian Kummer extensions: Image transcript and my attempts are below: Let $K/F$ be a Galois extension with Galois group $G=\operatorname{Gal}(K/F)$ of order $n$. ...
hbghlyj's user avatar
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Understanding a proof in JS Milne's Fields and Galois Theory (Prop 7.10)

The following is proposition 7.10 in Milne's Fields and Galois Theory: Let $G$ be a group of automorphisms of a field $E$, and let $F=E^G$ (ie. $F$ is fixed field of $G$). If $G$ is compact and the ...
Ajin Shaji Jose's user avatar
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Why the Galois group of polynomial $p(x)=x(x^4+12x^3+11x^2+11x+13)$ over $\mathbb Q$ can't be $S_5$ or $A_5$?

Why the Galois group of polynomial $p(x)=x(x^4+12x^3+11x^2+11x+13)$ over $\mathbb Q$ can't be $S_5$ or $A_5$? I know that The Galois group of an irreducible polynomial over $\mathbb Q$ is transitive ...
Fuat Ray's user avatar
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why the map $X \mapsto X^2$ induces a $K$-monomorphism?

Suppose that $L: K $is finite and every $K-$monomorphism $L \to L$ is an automorphism. Does this result hold if the extension is not finite? I found the answer here It written that It is necessary ...
jasmine's user avatar
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Galois group of splitting field of $x^3-5$ over $\mathbb F _7$

Honestly, I'm not even sure where to start. I think I understand how to find the Galois group of a field extension with $\textrm{char}\mathbb F=0$ but for some reason I'm confused when it comes to ...
RatherAmusing's user avatar
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1 answer
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What are good sources for examples of field extensions?

I need to understand field extensions and Galois theory. I have read about the theory. But I think to digest the subject, i need to see lots of (non)normal, (in)separable, (in)finite dimensional, (non)...
boyler's user avatar
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Is every algebraic extension of a finite field Galois?

Let $E/F$ be a (not necessarily finite) algebraic extension, where $F$ is finite. Now, it is known that $E/F$ is a normal extension. On the other hand, $E/F_p$ is algebraic and, since $F_p$ is perfect,...
A Name's user avatar
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In finite fields, generators of $F^*$ under automorphisms in Galois group are also generators

Let $F$ be a finite field with characteristic $p$ and denote $F^*$ the group of invertible elements of $F$. Show that if $a \in F^*$ is a generator, then so is $\sigma (a)$, for all $\sigma \in \...
RatherAmusing's user avatar
4 votes
1 answer
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Searching Explanation/References for a certain Corollary to Chebotarev Density Theorem

I am reading “Notes on the model theory of finite and pseudo-finite fields” by Zoé Chatzidakis https://www.math.ens.psl.eu/~zchatzid/papiers/Helsinki.pdf and on page 16 (or in this newer version of ...
Dainka's user avatar
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The Frobenius Endomorphism is Surjective iff the field is perfect

I'm taking a Galois theory class right now. I've read and understood the proof that the Frobenius endomorphism is surjective iff the field is perfect (working in characteristic $p$). But it just feels ...
Boran Erol's user avatar
3 votes
1 answer
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Constructing a primitive element for each fixed field of a Galois extension

Let $K/F$ be a finite Galois extension with Galois group $G$ and normal basis $ \{\sigma(\alpha):\sigma\in G\} $ and let $H\leq G$ be a subgroup. I am asked to show that that $F(Tr_{K/K^H}(\alpha))=K^...
A Name's user avatar
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1 answer
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Isomorphism of topological groups

Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
ByteBlitzer's user avatar
2 votes
1 answer
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Show Galois Group of a polynomial is isomorphic to $S_n$

We just moved on from Galois theory to solving polynomial equations and I don't quite understand a problem we were given: Let $p \geq 5$ be prime and $0 < a_1 < ... < a_{p-2}<b$ be even ...
Very Interesting's user avatar
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Compute the order of group generated by $X \mapsto X+1$ and $X \mapsto\frac{1}{X}$ which are in $\operatorname{Aut}(\mathbb{F}_5 (X) / \mathbb{F}_5 )$

The exercise is as follows: Let $\sigma$, $\tau \in Aut(\mathbb{F}_5 (X) / \mathbb{F}_5 )$, where $\sigma (X) = X+1$ and $\tau (X) = \frac{1}{X}$. ($X$ is the symbol or variable.) Let H be the group ...
shwsq's user avatar
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There exists a transitive group action of $S_n$ on $n!$ elements.

I don't believe this statement, so I think I'm making a mistake. I've asked ~10 people, and nobody found a mistake in the proof. Here it is. Let $L/K$ be a Galois extension with Galois group $S_n$. (...
Boran Erol's user avatar
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1 answer
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I don't understand the proof that extensions that aren't separable have 0 trace in Keith Conrad's norm and trace notes.

I don't understand the proof that extensions that aren't separable have $0$ trace in Keith Conrad's norm and trace notes. I don't understand the following argument. This argument is used while ...
Boran Erol's user avatar
2 votes
2 answers
128 views

Show that $\mathbb{Q}(\sqrt[3]{5} \cdot \sqrt{2}) \subset \mathbb{Q}(\sqrt[3]{5}+ \sqrt{2})$

How can I show that the extension $\mathbb{Q}(\sqrt[3]{5} \cdot \sqrt{2})$ is contained in the extension $\mathbb{Q}(\sqrt[3]{5} + \sqrt{2})$? I need to show that $\sqrt[3]{5} \cdot \sqrt{2}$ can be ...
lkksn's user avatar
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Prove that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}, \xi_3,\xi_5) \colon \mathbb{Q}] = 240$.

I have already proven that $[\mathbb{Q}(\sqrt{3}, \sqrt[3]{3}, \sqrt[5]{3}) \colon \mathbb{Q}] = 30$ and $\sqrt{3} \not\in \mathbb{Q}(\xi_5)$. Consequently, $[\mathbb{Q}(\xi_5,\sqrt{3}) \colon \...
David's user avatar
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Why does $x^3-7$ have Galois group isomorphic to $S_3$? [duplicate]

I'm not concerned with showing that the order of the Galois group is $6$; I've already done that. I'm more concerned with the structure of the Galois group. So $x^3-7$ has the roots \begin{equation} ...
Grigor Hakobyan's user avatar
2 votes
0 answers
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Cohomology class of automorphism group of Galois form

Let $\Gamma$ be the Galois group of a finite Galois extension $K/k$ of fields of characteristic zero. Let $G$ be an algebraic group defined over $k$. Let $G'$ be another algebraic group over $k$. We ...
gimothytowers's user avatar
27 votes
1 answer
1k views

"Galois theory" on graphs

Let $G = (V, E)$ be a graph. For the sake of simplicity, let us assume $G$ is undirected and finite. The automorphism group $\mathrm{Aut}(G)$ contains all graph isomorphisms $\phi : G \rightarrow G$. ...
safsom's user avatar
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2 votes
1 answer
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Subfields of splitting field of $x^4+25$ over $ℚ$.

Let $F$ be the splitting field of the polynomial $x^4+25$ over $ℚ$. List all subfields in $F$ and the corresponding subgroups in the Galois group. is problem $1$ on this pdf. The solution is: As we ...
hbghlyj's user avatar
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3 votes
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Transforming the reduced sextic $x^6+x^2+ax+b$ into a quintic

The Bring-Jerrard quintic: $$x^5+x = a$$ For $a≠0$. After some change we get: $$\frac x a = a^{-4/5}\left(1- \frac x a\right)^{1/5}$$ Expanding the RHS using binomial theorem because $\frac x a = \...
Thinh Dinh's user avatar
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0 answers
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Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$

So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
NoetherBoy 's user avatar
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Why is this extension Galois?

I'm reading a proof of the FTA (taken in the book Algebra : Chapter 0 by Paolo Aluffi) and I'm having a hard time understanding the beginning : "Let $f(x) \in \mathbb{C}[x]$ be a nonconstant ...
Ceru's user avatar
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2 votes
1 answer
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If $f(x)\in \mathbb{Z}[x]$ is irreducible (over $\mathbb{Q}$), is it always possible to find $a$ and $b$ in $\mathbb{Q}$ with $f(ax+b)$ Eisenstein? [duplicate]

My initial thought is no, simply because it seems too easy if it is true. The simplest example of a nontrivial irreducible polynomial I could think of was $f(x)=x^2+1$. Unfortunately, $f(x+1)$ is ...
ljfirth's user avatar
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uniquness of finite fields if they are inbedded in a algebraic closure

I read in the book of Bosch 3.8 after Theorem 2, that if we fix a algebraic closure of $F_p$ all fields of char p with q elements are equal (not just by isomorphism but really equal). His argument is ...
user1072285's user avatar
1 vote
1 answer
41 views

Does the Galois group of a polynomial change upon translation?

I know that a polynomial $f(X)$ is irreducible iff $f(X+1)$ is irreducible. Is it true for the Galois group? I think it should but I don't know if there ir a neat proof (I have thought of writing the ...
Valere's user avatar
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2 votes
1 answer
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Artin-Schreier extension is cyclic of degree 1 or $p$

Let $K$ be a field of characteristic $p > 0$ and $K \subset L$ the extension obtained by adjoining the zeros of the Artin–Schreier polynomial $f = x^p − x − a \in K[x]$, where $a\in K^*$, to $K$. ...
math_physics's user avatar
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1 answer
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Why is the subfield of $\mathbb{Q}(\zeta_p)$ of index $2$ expressible in terms of the sum of $\zeta_p$ to the power of all quadratic residues mod $p$?

Let $p$ be an odd prime, $\zeta_p = e^{2 \pi i / p}$. I've been playing around with calculating the intermediate fields of $\mathbb{Q}(\zeta_p)$. I know that $\mathrm{Gal}(\mathbb{Q}(\zeta_p) / \...
Robin's user avatar
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3 votes
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Is a sphere, whose volume is a sum of volumes of two given constructible spheres, constructible?

If I have two spheres with radius $r_1$and $r_2$ a sphere with the volume as the sum of volumes of these two spheres will have radius $(r_{1}^3 +r_{2}^3)^{1/3}$ now given that $[\mathbb{Q}(r_1):\...
Anurenj ER's user avatar
3 votes
0 answers
55 views

Detecting isomorphism of simple field extensions

Let $k$ be a field and $f, g \in k[x]$ irreducible. How can I tell whether the field extensions $k[x] / (f)$ and $k[x] / (g)$ are isomorphic as extensions of $k$? Is there any necessary and sufficient ...
Tobias Fritz's user avatar
2 votes
1 answer
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$X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$

I'm asked to prove that $f(X)=X^{6}+(T-22) X^{4}+T X^{3}+(T-22) X^{2}+1$ is irreducible over $\mathbb Q(T)$. For this, I need first to prove that the polynomials $p(X) = X^{6}-22 X^{4}-22 X^{2}+1$ and ...
mathcounterexamples.net's user avatar
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Artin's theorem exercise - proving that the fixed field is generated by the coefficients of the minimal polynomial

Suppose $L/K$ is a finite extension. $G$ is a finite group of $K$-automorphisms of $L$. Denote by $L^G$ the field elements of $L$ fixed by action of $G$. For any $\alpha \in L$ we write $f(t, \alpha) =...
Featherball's user avatar
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1 answer
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showing that $x^4+x^3+2$ is primitive over $\Bbb F_3$

I want to show that $x^4+x^3+2$ is primitive over $\mathbb{F}_3$. By definition, this means that $x^4+x^3+2$ is monic and has a root $\alpha$ that generates the multiplicative group of $\mathbb{F}_{3^...
doctor's user avatar
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2 votes
1 answer
89 views

When can we divide a shape into n equal parts (like Gauss and Abel)?

Gauss' theorem on the constructibility of the regular n-gon is very famous. I recently came across the similar theorem by Abel on dividing the lemniscate into n equal parts, and wondered if there are ...
ljfirth's user avatar
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0 answers
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Computation of Norm

I am attempting an exercise from a Galois Theory problem sheet I found online. It asks: Let $p$ be an odd prime. Let $L_1 = \mathbb{Q}_p(\zeta_p) / \mathbb{Q}_p$ and let $L_2=\mathbb{Q}_p(\sqrt[p-1]{-...
Todd Burnett's user avatar
8 votes
1 answer
152 views

Assistance with an exercise on field endomorphisms

Working through the problems in a book on field theory (Field Extensions and Galois Theory by Bastida). I came across one which I thought looked like a "routine" exercise, but has been ...
Matt D's user avatar
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0 answers
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Automorphism group of $x^3 - x - y^2$.

Let $k$ be some field of characteristic $\neq 2$. The polynomial $f(x,y) = x^3 - x - y^2$ is irreducible in $k[x,y] \cong k[x][y]$ as it cannot have a root in $k[x]$. By Gauss's lemma, $f$ is then ...
Lisa's user avatar
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0 answers
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Automorphism group gives the Galois group over the fixed field.

Question. Let $E$ be a field and let $G=\operatorname{Aut}(E)$ be the group of ring automorphisms. Let $F=\operatorname{Fix}(G)$ be the fixed field of $G$. Is $E$ Galois over $F$? Is $G=\operatorname{...
user108580's user avatar
2 votes
0 answers
55 views

For every integer $r>1$, there is a Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z} / r \mathbb{Z}$. [duplicate]

I want to show that for every integer $r>1$, there is a Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z} / r \mathbb{Z}$. I'm not really sure how to do this - I know that if $p$ is ...
Robin's user avatar
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1 vote
0 answers
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Restrictions of Galois representations

What field embeddings do I need to fix so that if I have a Galois representation $\rho:G_F \to GL(V)$, where $F$ is a number field, then for all finite extensions $L/F$ and places $v$ of F, I can talk ...
user14411's user avatar
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2 votes
0 answers
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Patrick Morandi "Field and Galois Theory" - Exercise I.3.12

From the book: Let $K$ be a field, and suppose that $\sigma \in \mathrm{Aut}(K)$ has infinite order. Let $F$ be the fixed field of $\sigma$. If $K / F$ is algebraic, show that $K$ is normal over $F$. ...
Yifan Dai's user avatar

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