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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Is $K := \mathbb{Q}(\cos (2\pi / 11))$ a Galois extension over $\mathbb{Q}$?

I believe that it is because $\cos(2\pi / 11) = (\zeta + \zeta^{-1})/2$ where $\zeta = e^{2\pi i/11}$ is a primitive $11$-th root of unity, and so $K$ is a subfield of $\mathbb{Q}(\zeta)$ with ...
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11 views

Maximal unramified extension of a global function field

Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?
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1answer
19 views

Galois group of maximally tamily ramified extension over the maximally unramified extension of a global function field F

Suppose $F$ were a local nonarchimedean field of characteristic zero of residue characteristic $p$. Let $F^{un}$ the maximally unramified extension. Let $F^{un}\subset F^{tame}$ be the maximally tame ...
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0answers
22 views

Field norm of $F(\sqrt[n]{a})$

Let $F$ be a field of characteristic zero that contains a primitive $n^{th}$ root of unity. Pick $a$ such that $K=F(\sqrt[n]{a})$ is a cyclic extension of $F$ of degree $n$. Let $\sigma$ be a ...
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22 views

Question about the index of a subgroup in $\mathrm{Aut}(\mathbb{C} / K )$ with $K$ a number field.

Suppose that $k_0$ is a number field with subfield $K$. Set $[k_0 : K] = d$. If $G = \mathrm{Aut}(\mathbb{C} / K )$ and $H$ is the subgroup of $G$ which fixes $k_0$, is it true that $[G:H] = d$? ...
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38 views

Splitting even degree polynomials

I have an octic equation (degree $8$) and a sextic equation (degree $6$) in $\Bbb Z[x]$ with very large coefficients (size several hundred bits) that I know splits into two quartics and two cubics ...
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1answer
20 views

How to split a quartic into two quadratics?

I have a quartic in $\Bbb Z[x]$ with very large coefficients that I know splits into two quadratics in $\Bbb Z[x]$. What is the best way to do find the quadratics?
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1answer
99 views

Solve $x^3+x+3=0$ by Galois's theory

Solve with radicals the following equation $x^3+x+3=0$, using Galois Theory. I'm just starting learning this and I do not have many ideas.
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1answer
46 views

Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
2
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1answer
27 views

Basis for the field extension $\mathbb{Q}(\zeta_{12})$

Consider the cyclotomic field $\mathbb{Q}(\zeta_{12})$ where $\zeta_{12}$ represents the $12$-th primitive root of unity. Since the minimal polynomial of $\zeta_{12}$ is given by $\Phi_{12}(x)$ which ...
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1answer
21 views

Why the action of $\mathrm{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)$ on $\overline{\mathbb Q}_p$ restricts to $\overline{\mathbb Q}$?

Let $\overline{\mathbb Q}$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$ and chose an algebraic closure $\overline{\mathbb Q}_p$ for $\mathbb Q_p$. The embedding $\mathbb Q \hookrightarrow ...
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2answers
61 views

The algebraic closure of $\mathbb{Q}_p$

I am trying to explain why $\mathbb{Q}_p^{\text{alg cl}}$ is an infinite field extension of $\mathbb{Q}_p$ (unlike $\mathbb{C}/\mathbb{R}$ which has deg 2). Does the following argument work out... ...
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54 views

Galois group over $\mathbb{Q}$ [closed]

Let $$\begin{align*} K&=\mathbb{Q}(\{\text{all $2^n$-th roots of unity for $n\in\mathbb{N}$}\})\\ L&=\mathbb{Q}(\{\text{all $n$-th roots of unity for $n\in\mathbb{N}$}\}) \end{align*}$$ What ...
2
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1answer
36 views

Relation between roots of an irreducible polynomial

Let $K$ be a field, and $\alpha$ be an element of a separable extension of $K$, such that $\alpha^p\in K$, but $\alpha\notin K$, $p$ a prime. Let $f$ be the minimal polynomial of $\alpha$ over $K$ ...
3
votes
2answers
56 views

When does a Galois group of a quintic have order divisible by three?

Apparently, nice necessary and sufficient conditions are known for a Galois group of a degree 5 polynomial to have order divisible by 3. What are these conditions? The possible Galois groups for an ...
2
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1answer
47 views

Problem involving cubic field extensions

Let $F$ be a field of characteristic $0$ and let $L$ be a cubic extension. I want to show that there exists an element $a \in F,$ and an extension $L_0$ of $\mathbb{Q}(a)$ such that ...
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2answers
58 views

Problem on Galois theory and irreducible polynomial

Let $p,q$ be primes, estimate the degree $[\Bbb Q(\sqrt[p]{2}\cdot\sqrt[q]{2}):\Bbb Q]$ and prove that the polynomial $X^q-2$ is irreducible in the ring $\Bbb Q(\sqrt[p]{2})[X]$ I found this problem ...
2
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1answer
34 views

Why is the degree of this field extension $[K(x,y): K(x^p, y^p)]= p^2$?

Fix a prime $p$. Let $K:=\overline{\mathbb{F}}_p$ be the algebraic closure of $\mathbb{F}_p$. Consider now the field $K(x,y)$ of rational functions in $x,y$ and its subfield $K(x^p,y^p)$ of rational ...
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1answer
50 views

Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$

Suppose $L=\mathbb{Q}(\sqrt{a+b\sqrt{d}})$,($d$ and $a+b\sqrt{d}$ are square free algebraic integers) when is $L/\mathbb{Q}$ a normal extension? When does $Aut(L/\mathbb{Q})=\mathbb{Z}/4\mathbb{Z}$ or ...
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56 views

Analogy between Galois groups and fundamental groups

I've heard that there is an analogy between algebraic field extensions and covers (in topology). In this analogy Galois extensions correspond to Galois covers and Galois groups correspond to ...
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1answer
54 views

Why the extension dimension of $x^3-2$ equal to $6$?

I have seen couple questions related to this one, but after reading the answers I am still confused: Why is the extension dimension of $x^3 - 2$ equal to $6$? In other words, why are the basis ...
2
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2answers
41 views

$K/F$ Galois Extension, $H \leq G$, where $G$ the Galois group, then there is $\alpha \in K$ s.t. $H=\{\sigma \in G : \sigma \alpha = \alpha\}$.

Let $K/F$ be a finite Galois extension of fields with Galois group $G$. Let $H$ be a subgroup of $G$. Then there is $\alpha \in K$ such that $H=\{\sigma \in G : \sigma \alpha = \alpha\}$. My proof ...
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1answer
32 views

Distribution of the sumset of two GF($q$) subsets

First, a simple definition. The sumset of two subsets $\mathcal{S}_1$ and $\mathcal{S}_2$ containing $GF(q)$ elements is defined as: $$\mathcal{S}_1 + \mathcal{S}_2 = \left\{ s_1 + s_2:s_1 \in ...
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1answer
29 views

Galois Group of the splitting field of the polynomial of $x^{11} - 7$ over $\mathbb{Q}$

Be $\mathbb{L}$ the splitting field, and be $G$ the Galois Group of $\mathbb{L}/\mathbb{Q}$, I've to prove that $G \cong A \ltimes B $ with $|A| = 11$ (cyclic) and $|B|=10$, abelian. The central ...
2
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1answer
33 views

Reference/advice for infinite Kummer extension $\mathbb{Q}_p(\sqrt[p^{\infty}]{p}, \zeta_{p^{\infty}})$

I am looking for help with a good reference for infinite Kummer extensions, or a check if I am on the right lines. In particular I need to refer to the Galois group of the field extension ...
3
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1answer
65 views

Finding the Fixed Fields in the Galois Correspondence for the Splitting Field of $x^4-3x^2+4$ over $\mathbb{Q}$

I have found the Galois group of the polynomial $x^4 - 3x^2 + 4$ (see below), but I am not sure how to find the fixed fields in the Galois correspondence. The roots of the polynomial are $$\pm ...
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24 views

Question on primitive element for extensions of finite fields [duplicate]

I was given this question in field theory class which I am stumped on as after some effort I still cannot tackle. I am given a general natural number $n>1$ and am asked to prove or disprove: ...
4
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1answer
35 views

splitting field of $x^8-1$ over $\mathbb F_3$

Suppose $F=\mathbb F_3$ and $f(x)=x^8-1$ in $F[x]$. I tried finding the Galois group of the splitting field of $f(x)$ over $F$ and I'm not so sure if what I did was correct. I began by looking at ...
4
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2answers
51 views

primitive element $a$ of $\mathbb F_{p^n}/\mathbb F_p$ such that $a^n\in\mathbb F_p$

Is it true that for every $n\in \mathbb N$ there exists a prime $p$ such that the extension $\mathbb F_{p^n}/\mathbb F_p$ has a primitive element $a\in \mathbb F_{p^n}$ and $a^n\in\mathbb F_p$? I ...
4
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2answers
49 views

Showing that the Field Extension $\mathbb{Q}(T^{1/4})/ \mathbb{Q}(T)$ is not Galois

Prove that $\mathbb{Q}(T^{1/4})$ is not Galois over $\mathbb{Q}(T)$, where $T$ is an indeterminate. I am not sure how to proceed due to the indeterminate. It suffices to show that the degree of ...
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1answer
38 views

Abel-Ruffini implications

There are similar questions already regarding this topic but I'm hoping the answers can be diluted for me here. So regarding the fact that the roots of 5th order polynomials cannot be expressed as ...
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2answers
35 views

Galois group and degree of splitting field over complex rational functions.

Suppose $F=\mathbb C (t)$ the field of rational functions over $\mathbb C$. let \begin{equation*}f(x)=x^6-t^2\in F[x]\end{equation*}Denote $K$ as the spliting field of $f$ over $F$. I'm trying to ...
2
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2answers
47 views

The Galois group of a specific polynomial $ f(x) = x^6-2x^3+2 \in Q[x] $

Hello all I was given this question in Algebra asking me to show a polynomial's Galois group has both a subgroup and quotient group isomorphic to $S_3$ the three symmetric group. The polynomial in ...
3
votes
1answer
37 views

Galois conjugation in $\mathbb{Z}/m\mathbb{Z}$

In Silverman's Arithmetic of Elliptic Curves, he introduces the Weil pairing as a means of making the determinant pairing Galois invariant. He writes that $\det(P^{\sigma},Q^{\sigma})$ and ...
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1answer
31 views

Universal mapping property and field extensions

The following is a portion of notes for an introduction to Galois theory (I am not in the course): There are possible details missing but I can't find a similar discussion using the universal ...
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3answers
42 views

Galois group vs Permutation subgroups.[Confusion]

Okay my main problem rest with this quote from Rotmans group theory: Not every permutation of the roots of a polynomial $f(x)$ need correspond to some $\sigma\in Gal(K/F)$ but then he uses the ...
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Uses for coefficients other than the trace and norm for an element belonging to a field.

The trace and norm are very useful as maps from a field extension to the base field since they are multiplicative/additive and have a lot of other nice properties. They can be defined as two of the ...
3
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1answer
42 views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Problem: Let $K(\alpha)/K$ and $K(\beta)/K$ algebraic field extensions so that their respective Galois groups are abelian. Prove that the Galois group of the field extension $K(\alpha + ...
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47 views

Filters, nets and Galois correspondence

In the lecture, our prof. mentioned that the correspondence between nets and filters is a Galois correspondence without giving any more details about that. In algebra, the proof of the Galois ...
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$A_4$ extension of $\mathbb{Q}$ ramified at one prime

How can one show that an $A_4$-extension of $\mathbb{Q}$ ramified at only one prime must be totally real?
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1answer
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“If $\text{Gal}\left(K/F\right)=\left<\sigma\right>$, $N_{K/F}\left(\alpha\right)=1$ , then $\alpha=\frac{\beta}{\sigma\left(\beta\right)}$.”

For (fintie) Galois extension $K/F$, it is easy to show that $N_{K/F}\left(\frac{\beta}{\sigma\left(\beta\right)}\right)=1$ for all $0\ne\beta\in K$, $\sigma\in\text{Gal}\left(K/F\right)$. I want to ...
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43 views

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

I want to find dim. $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$ for positive rational number $a$ and $n^\text{th}$ root of unity $\zeta_{n}$ with assumption ...
11
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2answers
218 views

Galois Group of $x^{4}+7$

I am trying to find the Galois group of $x^{4}+7$ over $\mathbb{Q}$ and all of the intermediate extensions between $\mathbb{Q}$ and the splitting field of this polynomial. I have found that the ...
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2answers
59 views

Why normal subgroup chains in Galois theory

I have began understanding Galois theory and I had a question regarding the relationships of normal subgroups to field extensions. So given an irreducible polynomial over the rationals $$a_1 + a_2x ...
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75 views

Does such a Galois extension exist?

Let $K = \mathbb{Q}(\sqrt{-3})$, an imaginary quadratic field. Does there exist a finite Galois extension $L/\mathbb{Q}$ which contains $K$ such that $Gal(L/\mathbb{Q})$ is isomorphic to $S_3$? Here ...
11
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1answer
444 views

Sum of roots of cubic = -coefficient of quadratic term?

Working through Ian Stewart's "Galois Theory, Third Edition," he states at the end of the second paragraph on page 13: "Because we know that $\alpha_1+\alpha_2+\alpha_3$ is minus the coefficient of ...
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1answer
41 views

Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields? The sources on CFT that I have at hand only deal with CFT in general and then proceed ...
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1answer
41 views

Confusions about Galois theory and the quintic

I am very confused about Galois theory, because I have never seen a rigorous presentation of it. In particular, I don't understand what the books mean by "radical expression". It is also not clear ...
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68 views

Equation of 27 lines on a cubic surface [closed]

For a smooth cubic surface $S$ in $\mathbf{P}^3$, there're always $27$ lines on it, with the same configuration. We know the automorphism group of the lines is not solvable. How do we show the ...
7
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1answer
103 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...