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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Quartic Equation having Galois Group as $S_4$

Suppose $f(x)\in \mathbb{Z}[x]$ be an irreducible Quartic polynomial with Galois Group as $S_4$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$.Now, the Question is: Prove that $K$ ...
9
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1answer
236 views

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

This is the problem I am facing: Compute the intermediate fields of the extension $K | \mathbb{Q}(i)$ where $K = \mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i)$ and find the intermediate fields $M$ such ...
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1answer
51 views

Learning Galois theory - required subtopics that are prerequisite?

This is not a reference request, that is, I have access to many textbooks I am happy with. What I don't know is, what are the things I need to know to get started? My idea on the path of knowledge ...
4
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1answer
74 views

Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can understand normal subgroups). I read the ...
12
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1answer
317 views

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 ...
36
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3answers
9k views

How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
3
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0answers
33 views

Can this cyclic septic be solved using only one 7th root extraction?

I. Quintics For an example of a cyclic quintic, we have for $p=11$, $$x^5 +x^4 −4x^3 −3x^2 +3x+1 = 0\tag1$$ The five roots $x_k$ for $k=0,1,2,3,4$, in radicals, are, $$x_k = ...
4
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2answers
82 views

Is there any irreducible polynomial over $\mathbb{Q}$ whose Galois group is $S_4$?

As simple as that: Can we find an irreducible polynomial in $\mathbb{Q}$ such that, if $K$ is its splitting field over $\mathbb{Q}$, $\operatorname{Gal}(K|\mathbb{Q})\cong S_4$? I've thought a lot ...
5
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1answer
60 views

Finding a quartic polynomial in $\mathbb{Q}[X]$ with four real roots such that Galois group is ${S_4}$.

Is there a quartic polynomial $p(x)\in\mathbb{Q}[x]$ irreducible with four real roots such that Galois group is ${S_4}$? If it really exists, can someone give me a example?
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0answers
8 views

Finite extension of a closed differential field is closed.

Let G be the differential Galois group of a differential field extension M of K. Then any finite dimensional extension of a closed intermediate differential field is closed. I can not realise this ...
2
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1answer
49 views

From where can I study more about Dickson polynomials?

I know some basic bits about this construction as to how they effect permutations of Galois fields. But I want to get some detailed understanding of them. Any references?
2
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1answer
47 views

Galois group and traslations by rational numbers.

Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$. Now my question: ...
1
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1answer
22 views

Isomorphism of quadratic extensions (of a number field)

I think we agree that two (squarefree) quadratic extensions of $\mathbb Q$, say $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$ are not isomorphic. Now consider the following tower of fields ...
10
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3answers
214 views

Solve this tough fifth degree equation.

$$x^5+x^4-12x^3-21x^2+x+5=0$$ I think it can be solved by trigonometric ways but how?
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2answers
1k views

When the group of automorphisms of an extension of fields acts transitively

Let $F$ be a field, $f(x)$ a non-constant polynomial, $E$ the splitting field of $f$ over $F$, $G=\mathrm{Aut}_F\;E$. How can I prove that $G$ acts transitively on the roots of $f$ if and only if $f$ ...
1
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0answers
23 views

A “non-degenerate pairing” between $\operatorname{Gal}(K/k)$ and $K/k$

In this post, I'd like to compare Galois theory and homology theory. Due to the limit of my knowledge, I'm not sure if my consideration is right. I hope you can show me the right way. In topology, ...
0
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0answers
27 views

Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ .

Let $E/F$ be a finite extension and it is a Galois extension. Prove that the number of automorphisms from $E$ to $E$ that fixes $F$ is equal to $[E:F]$ . I cant start at all.How should I begin?
2
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1answer
20 views

Is a Galois group of a number field faithfully represented by its action on the set prime ideals of the ring of integers?

Is a Galois group $G$ of a number field $K$ faithfully represented by its action on the set prime ideals of the ring of integers $O_K$? This is true in some cases, like $Z[i]$. (Where we can see the ...
2
votes
1answer
89 views

Ring of integers of a cyclotomic number field

Let $\omega$ be the primitive $n^{th}$ root of unity. Consider the number field $\mathbb Q(\omega)$. How to show that the ring of integers for this field is $\mathbb Z(\omega)$? Also, find the ...
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1answer
241 views

What is a good way to find an algebraic field extension that is not separable and not normal?

1) I know K/F is normal if K is a splitting field of some S subset F. 2) K/F is a separable extension if every element of K is separable. ...but am having problems coming up with an extension that ...
3
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0answers
41 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
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1answer
45 views

Cyclotomic field over $\Bbb Q$

Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the ...
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0answers
18 views

How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
34
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1answer
990 views

Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such ...
2
votes
1answer
35 views

Number of Galois conjugates

Let $L/\mathbb Q$ be a (finite) Galois extension of degree $n$ with Galois group $\Gamma$. We know that there is a primitive element or generator $\alpha$ of this extension. My question: Is the ...
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1answer
66 views

Galois Group of $x^4 - x^2 - 3$

Find the Galois Group of $x^4 - x^2 - 3$ This is a qual question. I don't know how to find the splitting field of this polynomial.
2
votes
1answer
31 views

Cyclic extension without primitive root of unity

Let $F$ be a field that doesn't contain a primitive fourth root of unity. Let $L = F(\sqrt a)$ for some $a \in F - F^2$ and let $K=L(\sqrt b)$ for some $b \in L - L^2$. If we have $N_{L/F}(b) ...
3
votes
1answer
36 views

A constructive method for finding a subfield $\mathbb{K}$ such that $Gal\left(\mathbb{C}\left(x\right)/\mathbb{K}\right)$ is isomorphic to $S_3$.

Let $\mathbb{C}\left(x\right)$ be the field of complex rationals functions. Find a subfield $\mathbb{K}$ of $\mathbb{C}\left(x\right)$ such that $Gal\left(\mathbb{C}\left(x\right)/\mathbb{K}\right)$ ...
1
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1answer
45 views

How does one construct the Galois field extension $GF((2^2)^3)$?

Looking at past exam question, one asks us to construct a Galois field extension $GF((2^2)^3)$ whenever the primitive irreducible polynomial $p(X) = X^3 + \alpha X^2 + \alpha X + \alpha \in ...
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0answers
30 views

Prove that the coefficients of a polynomial are in a finite field

I am trying to understand the proof of the following statement: Let $\mathscr{θ}$ be an algebraic element over the finite field $F$ and $\mathscr{θ=θ_1,θ_2 ... θ_n}$ be all the conjugate elementes of ...
4
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0answers
53 views

How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$?

$\mathbb{F}/\mathbb{Q}$ is a finite extension. Denote $\dim_{\mathbb{Q}}\mathbb{F}=n$. How (possibly) many subfields does $\mathbb{F}$ has that extend $\mathbb{Q}$? Thoughts: if ...
4
votes
2answers
57 views

Why is this Galois group abelian?

Consider the field extension $\mathbb Q(\zeta_3,\sqrt[3]2,\zeta_8)/\mathbb Q(\zeta_3)$, with intermediate fields $\mathbb Q(\zeta_3,\sqrt[3]2)$ and $\mathbb Q(\zeta_3,\zeta_8)$. Denote ...
3
votes
1answer
57 views

Discriminant of Polynomials (Galois Theory)

So I'm reading Dummit and Foote and they define the discriminant of $x_{1},...,x_{n}$ by $$D=\prod_{i<j}(x_{i}-x_{j})^2$$ and the discriminant of a polynomial to be the discriminant of the roots. ...
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2answers
2k views

How to solve fifth-degree equations by elliptic functions?

From F. Klein's books, It seems that one can find the roots of a quintic equation $$z^5+az^4+bz^3+cz^2+dz+e=0$$ (where $a,b,c,d,e\in\Bbb C$) by elliptic functions. How to get that?
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0answers
17 views

Q-automorphisms determind by associates to id-element?

Let's say you consider the Galois group of $G(\mathbb{Q[\sqrt{3},\sqrt{2},i]}/\mathbb{Q})$. This is just an example. Is it correct that the $\mathbb{Q}$-automorphisms is determined up to associates? ...
1
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1answer
43 views

Question from 14.6 “Galois Groups of Polynomials” from Dummit and Foote

I am confused in the proof of proposition 30 in Dummit and Foote on page 608. Near the end of this "proof" he goes on to say, By the Fundamental Theorem of Galois Theory, the fixed field of ...
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0answers
73 views

Determining the cycle type of complex conjugation

This arose recently in an online discussion about roots and irreducibility. Let $f(X) = X^4 - 4X + 2$. $f(X)$ has two real roots and two complex roots, which means that complex conjugation $\sigma$ ...
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25 views

Galois extension and Galois group

Let $x$ be a real root of the polynomial $X^3-X+1$, and $y,\overline{y}$ two other roots in $\mathbb{C}$, and $K$ be the cubic field $\mathbb{Q}[x]$. Show that $y+\overline{y}=-x$ , ...
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0answers
31 views

Galois group for the algebraic closure of a finite field

If $N$ is the algebraic closure of a finite field $F$, prove that $\operatorname{Gal}(N/F)$ is an abelian group and that any element of the Galois group has infinite order. If $N$ were a finite ...
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2answers
27 views

Can we simply say this regarding the number of elements in the Galois Group?

Consider a polynomial like $x^4-10x^2+1=0$, which has four distinct roots $\pm \sqrt{2} \pm \sqrt{3}$. The Galois Group has 4 elements, so the Galois Group is isomorphic to the Klein-4 group. Now ...
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0answers
21 views

Galois closure uniqueness confusion

I'm a little bit confused by the statement of this corollary, (Corollary 23 pg 594 of Dummit and Foote) Let $E/F$ be any finite separable extension. Then $E$ is contained in an extension $K$ which is ...
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0answers
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Find and show primitive element of field extension

I have the polynomial $\ f(x)= x^{15}-3 $ and want to find the splitting field. The splitting field is surely $\Bbb Q$ adjoined the roots of the polynomial. I want to write the splitting field as ...
3
votes
1answer
35 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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1answer
40 views

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

I have seen many textbooks state this result without proof. $``$ If $E$ is the splitting field for the polynomial $f=x^p-1 \in \mathbb{Q}[X]$ where $p$ is prime, then the Galois group ...
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1answer
49 views

Prove that $K$ is finite Galois over $\mathbb{Q}$

I just need a bit of quick help in understanding some solutions to a problem set. The question is this: (a) Let $K=\mathbb{Q}(\alpha)$ with $\alpha$ a zero of $f(x) = x^3-3x+1$. Prove that $K$ is ...
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0answers
40 views

The class number and the inverse Galois problem

Let $G$ be finite group and $k$ a field. Inverse Galois theory asks if there is a galois extension $L/k$ such that $Gal(L/k) \simeq G$. Lets assume $k=\mathbb{Q}$ and let $\mathcal{h}_L$ denote the ...
2
votes
5answers
606 views

Minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$

Is there an "easy" way to find the minimal polynomials of $\sin(\pi/8)$ and $\cos(\pi/9)$ without the help of any computer programme? If I knew $\sin(\pi/8)=\frac{\sqrt{2-\sqrt2}}{2}$ then it would ...
2
votes
0answers
26 views

Galois action on torsion points of formal group

My question is about a statement in Lang's Cyclotomic Fields, Ch. 8, $\S$2, although I've modified the notation a little. Let $R$ be a complete discrete valuation ring with fraction field $K$, ...
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1answer
93 views

To intermediate field is Galois$\iff$ Galois group over intermediate field is a normal subgroup

I completely edited my question and with the help of Lubin, I added my proof. Any comments are welcome. My question was: Let $F_0 \subset F_1 \subset F_2$ be fields, and suppose $F_2/F_0$ is a ...
0
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1answer
41 views

finding intermediate fields of field extensions: simple method vs. Galois theory

I am looking for a straightforward way to find all intermediate fields of a field extension. Let's take the splitting field of $X^3-2$ over $\Bbb Q$ as an example. If we adjoin the roots of $X^3-2$ ...