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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1answer
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How can I find all intermediate fields between $\mathbb{Q}\left(\sqrt[d]{a}\right)$ and $\mathbb{Q}$?

Let $a$ be a positive rational number, $n$ be a natural number, and $K$ be extension field of $\mathbb{Q}$ with $\sqrt[n]{a}$. I guess that intermediate field $E$ between $K$ and $\mathbb{Q}$ with ...
2
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0answers
42 views

What is $\operatorname{Gal}(\mathbb{Q (2^{1/3})}/\mathbb{Q})$?

What is $\operatorname{Gal}(\mathbb{Q (2^{1/3})}/\mathbb{Q})$? We implement the fact that every automorphism in the Galois group $\operatorname{Gal}(K/F)$ maps every root $\alpha \in K$ of $f(x) \in ...
0
votes
1answer
58 views

Roots of irreducible polynomial $1 + x + \cdots + x^4$

I'm studying some Galois theory and I'm trying to determine the Galois group of the extension $\mathbb{Q}(e^{2 \pi i/5}) \supset \mathbb{Q}.$ We consider the minimal polynomial of the element $e^{2 ...
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1answer
19 views

$[K:K\cap \mathbb{R}]$ for $K$ a Galois extension of $\mathbb{Q}$

Suppose $K$ is a Galois extension of $\mathbb{Q}$, the field of rational numbers. How do I prove that $[K:K\cap \mathbb{R}] \leq 2$, where $\mathbb{R}$ denotes the field of real numbers? I could ...
3
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1answer
35 views

Prove if $L = K(α_1, . . . , α_r)$ and each $\alpha_i$ is separable over $K$, then $L/K$ is separable

Let $L/K$ be a finite extension, $[L:K] = n$. Prove the following are equivalent: $L/K$ is separable $L=K(\alpha_1,...,\alpha_n)$ and every $\alpha_i$ is separable over $K$. I ...
1
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1answer
35 views

Difference Aut(F:K) and G(F:K)

What is the difference between the group of automorphisms that keep a subfield fixed versus the Galois group keeping the same field fixed? Are Aut{F:K} and G(F/K) just two ways of writing the same ...
1
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1answer
15 views

Question on the normal closure of a field extension

Hi all I was given the following question in my field theory class which I am stuck on: I am given $ K/F $ a finite extension of fields. I am asked to show the existence of an $ K \subset L $ such ...
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0answers
23 views

Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$

I am trying to find the Galois group of $(x^2-2)(x^3-2)(x^3-3)$ over the field $\mathbb{Q}(i\sqrt{3})$. The roots of this polynomial are $\pm \sqrt{2}$, $\zeta_3^k \sqrt[3]{2}$, and $\zeta_3^j ...
1
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1answer
22 views

Algebraic Closure terminology doubt

If F and K are fields, what does it mean when we say 'F is algebraically closed in K'?
3
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1answer
36 views

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd

Let $K$ be a field with $\mathbb{Q} \subset K \subset \mathbb{C}$ and $[K:\mathbb{Q}]$ odd. If $K/\mathbb{Q}$ is Galois, prove that $K$ is contained in $\mathbb{R}$. Find an extension with ...
1
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1answer
49 views

Why do we get exactly $2$ complex roots when $q > 10^5$?

I am working on the problem in my textbook: Construct a polynomial of degree $7$ with rational coefficients whose Galois group over $\mathbb{Q}$ is $\operatorname{Sym}(7).$ There is a theorem in ...
3
votes
1answer
37 views

Weil pairing, cyclotomic field in division field and determinant map for ell curve and abelian variety

Question 1: If $E/K$ is an elliptic curve defined over a number field $K$, then the Weil Pairing gives me that $K(\mu_n) \subseteq K(E[n](\bar{K}))$. If i identify $Gal(K(\mu_n)/K)$ with a subgroup ...
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0answers
9 views

Proof remainder of polynomial division GF(2) can be calculated by LSFR

I have been reading that CRC, which is the calculation of the remainder of $x/P(x)$ in GF(2) can be implemented with a Linear Shift Feedback Register. However, I can't find the proof for this, or ...
1
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1answer
18 views

Field Extensions of Q by radicals

Is Q(√6) = Q(√3,√2)? I understand that the degree of these field extensions comes from the degree of the minimal polynomial and (alternatively) basis of the field extension. I know that the basis of ...
2
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2answers
32 views

The galois group of a polynomial of degree 3 is either $A_3$ or $S_3$

Hungerford -Algebra p.271 Let $E/F$ be a Galois extension where $E$ is a splitting field for a separable irreducuble polynomial $f$ over $F$ whose roots are $a_1,a_2,a_3$. Let ...
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3answers
46 views

Galois extension of degree $ 2^n $

I'm trying to find a way to prove the following statement: Assume $ \mathbb{Q} \subset E $ is a Galois extension of degree $ 2^n $. Show that there are fields $ \mathbb{Q} = E_0 \subset E_1 \subset ...
1
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1answer
24 views

Galois group of $ x^n - a $ over a field containing $ \zeta_n $

I'm having trouble solving an exercise regarding Galois theory. Suppose $ n > 0 $ and $\zeta_n \in F \subset \mathbb{C} $, where $ \zeta_n $ denotes the primitive root of unity of degree $ n $. ...
3
votes
1answer
49 views

Galois group and intermediate fields for splitting field of $ x^3 -7 $

I'm trying to do the following exercise: find the Galois group $ G(E/\mathbb{Q}) $, where $ E $ is the splitting field of $ x^3 - 7 $, all its subgroups and the intermediate subfields $ E^H $ ...
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2answers
165 views

Galois extension of degree 75

I'm studying Galois Theory and I stumbled upon this question: Let $E/K$ be a Galois extension of degree $75$. Show that there exists an intermediate subfield $K \subsetneq F \subsetneq E$ such ...
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31 views

Kummer extensions, but then a bit more

I'm struggling with the following past exam question, parts (b) and (c): So, starting with (b): since $E/F$ is Galois of degree $p$, we know that $$\Gamma(E/F)\cong C_p\cong ...
3
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1answer
25 views

Construction of non-prime finite fields

I am new to Galois field theory and I am struggling with some definitions. To construct any non-prime finite field $GF(p^n)$ with p prime and $n \in \mathbb{N}$, one has to find an irreducible ...
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4answers
111 views

$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$

Prove that $$\mathbb{Q}\left ( \sqrt{2},\sqrt{3},\sqrt{5} \right )=\mathbb{Q}\left ( \sqrt{2}+\sqrt{3}+\sqrt{5} \right )$$ I proved for two elements, ex, $\mathbb{Q}\left ( \sqrt{2},\sqrt{3}\right ...
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1answer
32 views

Proof of a result on closed subgroups of Galois group

Let $M\supseteq K$ be an algebraic normal extension. Then its Galois group is profinite. I have been told that in this hypothesis, if $H\leq G$, then $H''=H\iff \bar H=H$, where ...
4
votes
3answers
83 views

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group

Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group with some elementary algebra, $x - \sqrt{2 +\sqrt{2}} = 0 ...
0
votes
1answer
36 views

Radical extension with root of cubic polynomial

If I take $f(x)$ is an irreducible cubic over $\mathbb{Q}$ with a root $\alpha$ in a splitting field and given that $\mathbb{Q}(\alpha)$ is a radical extension is it true that $\mathbb{Q}(\alpha) = ...
3
votes
1answer
40 views

Inertia group modulo $Q^2$

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
2
votes
3answers
83 views

If $a_1,a_2,a_3$ are roots $x^3+7x^2-8x+3,$ find the polynomial with roots $a_1^2,a_2^2,a_3^2$ [duplicate]

If $a_1,a_2,a_3$ are the roots of the cubic $x^3+7x^2-8x +3,$ find the cubic polynomial whose roots are: $a_1^2,a_2^2,a_3^2$ and the polynomial whose roots are $\frac{1}{a_1}, \frac{1}{a_2}, ...
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3answers
88 views

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$

Find the Galois group of the polynomial $x^{4} + x - 1$ over the finite field $\mathbb{F}_{3}$. I'm particurly struggling with what would be the approach to finding the splitting field of this ...
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0answers
29 views

Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $

I'm trying to solve the following problem: Let $ L =\mathbb{Q}(\varepsilon_5) $ be an extension of $ \mathbb{Q} $. Find the Galois group $ G(L / \mathbb{Q})$ $ \varepsilon_5 $ is the primitive root ...
1
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1answer
20 views

Decomposing the ring of integers of a number field as a sum of prime plus field fixed by ramification group

Let $L/K$ be a normal number field extension with ring of integers $\mathcal O_L/\mathcal O_K$. Let $Q$ be a prime ideal of $\mathcal O_L$ and inertia group $E = \{g \in ...
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0answers
42 views

How to find the Galois group of a given polynomial from Galois viewpoint?

Excerpt from Page 3 Consider the polynomial equation \begin{equation*} x^4 - 2 = 0, \end{equation*} with integer coefficients. We have four solutions: \begin{equation*} \alpha = ...
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3answers
61 views

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$

Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$ Let $\sigma$ be such that $\sigma(t)=-t$. I assume there is only one automorphism like this, I am not sure exactly why... How ...
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1answer
44 views

Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$

Let $\mathbb{K}=\mathbb{Q}(\sqrt[10]2)$. Show that $|\operatorname{Aut}(\mathbb{Q}(\sqrt[10]2))|=2$. Well, it is easy to see that the degree of this extension over $\mathbb{Q}$ is ten. Also, is ...
0
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1answer
28 views

What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
2
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1answer
83 views

Roots of $f(x) = x^3+x^2-2x-1$

Roots of $f(x) = x^3+x^2-2x-1$ Show: $a_1=2\cos(\frac{2\pi}{7})$ is a root of $f$. [Edited here] $a_2 = a_1^2-2$ is a root of $f$. $a_3 = a_1^3-3a_1$ is a root of $f$. The first one is ...
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0answers
65 views

Factorisation of large polynomials and Galois theory

As I understand it, one of the consequences of Galois theory is that there is no way of expressing the solutions to a general polynomial of degree 5 or higher in terms of radicals. Would a theory that ...
2
votes
1answer
25 views

On the expression of the Galois conjugates in terms of the coordinates in a basis

Let $K$ be a field and let $L$ be a Galois extension of $K$. Assume that $[L:K]=n$, and consider $e=(e_1, e_2, ...,e_n)$ a basis of $L$ over $K$. We note ...
2
votes
1answer
66 views

How to find minimal polynomial of primitive element (field theory)

I am given a primitive element $\alpha$ in the Galoisfield $F_{2^6}$. The question is to find the mimimal polynomial of $\alpha^7$. How to I find this? My thoughts so far: $$ \alpha^7 \rightarrow ...
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1answer
27 views

Irreducible Polynomials, and Galois Groups

I have a small question about some concepts. I know that if I have a polynomial $p(x)$ over a field $K$ and an extension field $F$, that elements of $\text{Gal}(F/K)$ permute the roots of $p(x)$ ...
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0answers
22 views

Why are inertia and decomposition groups only defined over normal extensions?

Can't we define them as the subgroups of the automorphism group of an arbitrary extension $L/K$ that fixes a prime $Q$ in $\mathcal O_L$ over a prime $P$ in $\mathcal O_K$? An ideal answer would ...
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0answers
40 views

What is the gcd of $x^3-2$ and $x^2+3ax+3a^2$?

I'm trying to understand Galois theory. $f(x)= x^3-2$, with $a$, $b$ and $c$ as roots. Galois resolvent $V_0= a+2b+3c$, $V_1= c+2a+3b$,... etc till $V_5=a+2c+3b$ assume $X=V_0$ ...
4
votes
1answer
60 views

Splitting field and galois group of $x^4-2x^2-1$ over $\Bbb{Q}$

I have that the roots of $m(x)=x^4-2x^2-1$ are plus minus $a=\sqrt{1+\sqrt{2}}$ and $b=\sqrt{1-\sqrt{2}}$ which is complex because $1-\sqrt{2}$ is negative. Taking out $i$ and letting $b=ic$ we have ...
2
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1answer
46 views

Extension of fields $[L:K]=2$. Show that there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$.

Let $K$ be a field with characteristic different from two, then if a extension of fields $L/K$ is such that $[L:K]=2$ then there is a element $\alpha\in L$ such that $L=K(\alpha)$ and $\alpha^2\in K$. ...
4
votes
2answers
49 views

Isomorphism type of the Galois group

$f=(x^2-2)(x^3-3)$. Let $K$ be the splitting field of $f$ over $\mathbb{Q}$. a) Determine the degree of extension of $K$ over $\mathbb{Q}$. b) Determine the isomorphism type of the Galois group of $K$ ...
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votes
1answer
15 views

Showing polynomial is irreducible over field containing roots of unity.

Given a field $F$ containing all the roots of unity I'm trying to show that $f(x) = x^p - \alpha^p$ is irreducible over $F$ (where $\alpha$ is not in $F$). It's clear that $f$ splits in $F(\alpha)$ ...
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0answers
34 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
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0answers
35 views

Determine the galois group of a quartic

I'm reading Hungerford's algebra chapter about galois theory. There is the following theorem in p.273 (with some minor changes) about determining the Galois group of a quartic: Let $K$ be a field ...
7
votes
1answer
75 views

Galois group of $(T^4-3)(T^6-3)$

Given the polynomial $f(T) = (T^4-3)(T^6-3)$, I would like to calculate the Galois group of $f$. What I've done is the following: setting $\alpha = 3^{1/4}$ and $\beta= 3^{1/6}$, $\zeta_k = e^{2\pi i ...
0
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0answers
21 views

Relations between galois group of polynomial and its factors

$f = f_1\dots f_n$ where $f_i$ is irreducible and distinct. What can i say about $\operatorname{Gal}(f/\mathbb{Q})$ if i know $\operatorname{Gal}(f_i/\mathbb{Q})$ for any(some) $i$.
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0answers
64 views

Galois group of $x^6-5x^3+6$

Let $f = x^6-5x^3+6$. I want to determine $\operatorname{Gal}(f/\mathbb{Q})$ without some group theory tricks (like Sylow's theorems) and without reduction $\bmod p$. Let $L_f = ...