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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Galois group of $x^4-2$

I am trying to explicitly compute the Galois group of $x^4-2$ over $\mathbb{Q}$. I found that the resolvent polynomial is reducible and the order of the Galois group is $8$ using the splitting field ...
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1answer
60 views

$\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ is a Galois extension, with $\mathbb{F}_4(x,y)$ a function field of an algebraic curve

Consider the field extension $\mathbb{F}_4(x,y) / \mathbb{F}_4(x)$ where $y$ is root of the polynomial $$ f(T)= x^4 + x^2T^2 + x^2T + x^2 + xT^2 + xT + T^4 + T^2 + 1 \in \mathbb{F}_4(x)[T]. $$ I ...
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Determine the elements of the Galois group [duplicate]

I want to determine the elements of the Galois group of $x^p-2$. I have never seen anything like this before and been struggling with some of the Galois problem. Thank you for any input!
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25 views

Way to check the Galois groups of given polynomials online?

Is there some way (for instance a way of typing into Wolfram Alpha) that will give me the Galois group of a given polynomial over Q.
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2answers
34 views

Generator of the cyclic Galois group $\operatorname{Gal}(\mathbb{Q}[\xi_{n}]/\mathbb{Q})$

I would like to know what is a generator of a $\operatorname{Gal}(\mathbb{Q}[\xi_{n}]/\mathbb{Q})$ cyclic group if we know that the elements of the group are automorphisms such as ...
4
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2answers
42 views

Example of Field Extension $E/F$ with $Char(F)=2$ and $[E:F]=2$, but is not Galois

I understand that for a field extension $E/F$, if $Char(F)\neq 2$ and $[E:F]=2$ then it must be a Galois Extension. I have proved this, but I am having trouble finding a counterexample when the ...
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1answer
26 views

General polynomial of degree $n$ is irreducible from Gauss' Lemma

I'm studying "old-fashioned" Galois theory and the following is an elementary and fundamental problem to keep proceed with my studies. I'm really stuck at that question. Can someone help me please? ...
4
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3answers
90 views

proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $

Here is an extract from my Galois Theory notes proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $ My question is after rearranging equation (1) has my lecturer omitted an ...
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0answers
34 views

All profinite groups are Galois groups (Thm 3.3.2 in J. Wilson's Profinite groups)

I am reading about infinite Galois theory in Wilson's Profinite groups, and I have a problem in understanding the proof of Lemma 3.3.1 and Theorem 3.3.2 (here you can see them). In particular, I don't ...
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1answer
22 views

Automorphisms of a field extension (proof verification)

I am asked to compute the automorphisms of the field extension $\mathbb{Q}(\sqrt[4]{2})/\mathbb{Q}(\sqrt{2})$. I know that $[\mathbb{Q}(\sqrt[4]{2}):\mathbb{Q}]=4$ since $$ ...
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2answers
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A question related to Galois theory

I'm working with this problem: Let L/K be a Galois extension with Galois group $S_4.$ Then L is the splitting field of a monic degree 4 irreducible polynomial over K. Char(K)=0. My method is since ...
3
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1answer
50 views

Why does this specific Quintic Equation have a closed form and this similar one does not?

I read on Wikipedia that x^5 -x -1 = 0 has a real root, but that you can't express it in radicals. So I thought maybe all of the x^5 -x -A =0 don't have a real root that can be expressed as a radical ...
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1answer
29 views

Cyclotomic Fields - Showing that the fixed field of $G(\mathbb Q(\xi)/\mathbb Q)$ is $\mathbb Q$.

If $p$ is a prime and $\xi$ is a primitive $p$th root of unity, I know that $G(\mathbb Q(\xi)/\mathbb Q) = \{\psi_{\xi,\xi^k}\}_{1\leq k<p}$, where for each $k$, $\psi_{\xi,\xi^k}(\xi) = \xi^k$. I ...
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0answers
29 views

Normal extension and action of automorphisms on factors

Let $N/K$ be a normal extension of fields. Let $f\in K[X]$ be an irreducible polynomial with monic irreducible factors $g,h\in N[X]$. Show that there exists an automorphism $\varphi$ on $N$ which ...
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3answers
37 views

How are all the roots of unity of cyclotomic extension are of this form? [closed]

Suppose $x \in Q(\zeta_n) $ which satisfy $x^t =1, t \in \mathbb{N}$. Then show that $x$ is of the form $\zeta_n^k$ for some $k$ where $1 \leq k \leq n-1$ ?
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0answers
14 views

Approximation of a trisecting an angle

I learned a proof that it is impossible to trisect an angle. Is there some research that if we have been given an angle, a ruler and a compass and we are allowed to draw $m$ circles and $n$ lines/line ...
2
votes
1answer
49 views

Determine whether the extension is Galois [duplicate]

I am trying to prove that $K=\mathbb{Q}(2^{1/3}, i\sin{2\pi/3})$ is Galois extension over $\mathbb{Q}$. It is easy to see that $K=\mathbb{Q}(2^{1/3},i\sqrt{3})$. I know it is Galois since $K$ is a ...
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3answers
43 views

If field $K/F$ is generated by the $\alpha_1,…,\alpha_n$, then an $\sigma\in $ Aut$(K/F)$ of $K$ uniquely determined

Does this proof seem correct? I'm having second doubts concerning the bolded material. Show that if the field $K$ is generated over $F$ by the elements $\alpha_1,...,\alpha_n$, then an automorphism ...
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0answers
42 views

Understanding algebraic closure [duplicate]

I am trying to understand what it means to have an extension that is an algebraic closure of the base field. I'm looking for someone who can help conceptually. I understand how $C/R$ looks. The basis ...
4
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0answers
48 views

What is the automorphism group of the field of all constructible numbers?

Let $\Omega\subseteq \mathbb{C}$ be the field of all constructible numbers (i.e. $\Omega$ is the smallest subfield of $\mathbb{C}$ which is closed under taking square roots). What is known about the ...
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1answer
31 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar ...
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1answer
47 views

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$.

Determine $n$, $m$ for which $\Bbb Q_n \subseteq \Bbb Q_m$. How to derive this. I am having no clue. I guess $n|m$. But what is the proof?
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27 views

A sequence of rational polynomials whose splitting fields over $\mathbf{Q}$ have dihedral Galois groups.

It is well known that the splitting fields of $x^3-2$ and $x^4-2$ over $\mathbf{Q}$ have Galois groups $D_6$ and $D_8$, Dihedral groups of $6$ and $8$ elements respectively. However, this pattern ...
2
votes
1answer
42 views

Galois theory, resolvent, Frobenius group

I need to prove that the Galois group of the polynomial $x^5+15x+12\in \mathbb{Q}[x]$ is the Frobenius group of order 20. The discriminant of that polynomial is $D=2^{10}\cdot 3^4\cdot 5^5$, i.e. it ...
0
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1answer
37 views

Let $G$ be the finite abelian group show that there is a galois extension $K/\Bbb Q$ with $Gal(K/\Bbb Q) \equiv G$ [duplicate]

Let $G$ be the finite abelian group show that there is a galois extension $K/ \Bbb Q$ with $Gal(K/\Bbb Q) \cong G$. I have seen one proof using For a fixed positive integer $n$, there are infinitely ...
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0answers
20 views

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$ [duplicate]

If $d \in \Bbb Q$ , show that $\Bbb Q( \sqrt d)$ lies in some cyclotomic extension of $\Bbb Q$. I have thaught that for $\Bbb Q(\sqrt 3)$ is in $\Bbb Q(w)$ where $w$ is in the $3$rd root of unity. ...
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53 views

Cyclotomic polynomial, after adjoining a radical

Suppose that $p>2$ is prime and that $a$ is a rational number for which $\sqrt[p]{a}$ is in $\mathbb C\backslash\mathbb Q$. The cyclotomic polynomial $\Phi_p$ is well-known to be irreducible over ...
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0answers
17 views

Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
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1answer
47 views

Galois group of $x^6+1$

$x^6+1$ has $6$ roots: $i,i\xi,i\xi^2,i\xi^3,i\xi^4,i\xi^5$ where $\xi=e^{\tfrac{2\pi i}{6}}$. Since $x^{12}-1=(x^6-1)(x^6+1)$ the splitting field of $x^{12}-1$ contains the splitting field of $x^6+1$ ...
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0answers
20 views

If $S=\sum (\frac{n}{p})\zeta^n$ then how to prove that $S^2=(\frac{-1}{p})p$? [duplicate]

Here $\zeta$ is a primitive $n$-th root of unity and ($\frac{n}{p}$) denotes the Legendre symbol. Can someone please give a proof of this fact? I tried writing $S^2$ as the product of two sums $S=\sum ...
4
votes
1answer
48 views

Prove the extension to be a Galois Extension

Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension. Trial: Since this $\mathbb C$ is a field of charactersitic $0$,it would be ...
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0answers
15 views

Finite Inseparable Extension

Preparing for my Galois theory exam in may and i am faced with the following question. Give an example of a finite inseparable extension with a sketched proof of its inseparability I have the ...
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1answer
41 views

Bases and Galois Theory

I am working through the following notes: http://www.win.tue.nl/~sterk/algebra3/hoofd.pdf I have come across Proposition 2.4.7 on Page 21 which is given without proof. For completeness and clarity, ...
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0answers
23 views

Does $[F(\zeta_n) : F ]$ divide $\phi(n)$?

I know that if $F=\mathbb Q$, the degree actually equals $\phi(n)$. Also, if the extension $F(\zeta_n)/F$ is Galois, then I can invoke my knowledge of the existence of an injective map from ...
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0answers
28 views

Show that K is a splitting field for some degree 4 polynomial f(x) in k[x].

Suppose $K/k$ is a finite Galois extension such that the Galois group of $K/k$ is isomorphic to $\mathcal S_4$. How can we show that $K$ is a splitting field for some degree $4$ polynomial $f(x)$ in ...
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1answer
28 views

$H$-orbits in X have not the same cardinality if $H$ is not normal in $G$

Let $G$ be a transitive subgroup of the symmetric group $S_n$ on $n$ letters, and let $H$ be a normal subgroup of $G$. I know that the action of $G$ on the set $X =\{ 1,..., n \}$ induces a natural ...
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1answer
42 views

Formula for quadratic equations from Galois theory

Can we deduce the classical formula for the solutions of a quadratic equations $ax^2+bx+c=0$ using Galois theory?
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19 views

Construct the rule in fields extensions

Let $f(x) = x^3+6x^2-12x+3$ Show that f(x) is irreducible over $\mathbb Q$ . Let $\theta$ be a real root of $f(x)$, which exists due to the intermediate value theorem. $\mathbb Q(\theta)$ consists of ...
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0answers
41 views

$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$?

I need to show that the extension $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ is Galois over $\mathbb{Q}$, and compute its Galois group. I am learning Galois theory by myself and got stuck in this exercise. I ...
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0answers
14 views

Show that invertible elements of the algebraic closure of $F_p$ is not cyclic

I want to show that invertible elements of the algebraic closure of $F_p$ is not cyclic, where p is a prime. I know that the algebraic closure of $F_p$ is countably infinite, since it is equal to the ...
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1answer
19 views

Sylow $p$-subgroup of a normal group.

Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. I need to show that any Sylow p-subgroup of $G$ is also contained in $H$. I know that any transitive ...
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2answers
19 views

Sylow $p$-group of a transitive subgroup of $S_p$

How do I show that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$? I am trying to prove a result of Galois and the only hint I have is that if $p$ is a ...
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0answers
22 views

Vector space multiplication matrix

I have to find normal basis for field $GF(3^6)$ which represents vector space over field $GF(3^2)$ and then find multiplication matrix for this vector space. Then I have to demonstrate fast ...
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1answer
64 views

Bourbaki's proof of normal basis theorem Part 2

Let $K/k$ be a finite Galois extension of a field $k$, $G$ its Galois group. The normal basis theorem states as follows. There exists an element $\alpha$ of $K$ such that $\{\sigma(\alpha)\ |\ ...
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1answer
19 views

Determine with proof the no of monic irreducible polynomials of prime degree $p$ over $F$,

Let $F$ be a field with $|F|=q$. Determine with proof the no of monic irreducible polynomials of prime degree $p$ over $F$, where $p$ need not be the characteristic of $F$. I know that $x^{p^n} -x$ ...
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votes
1answer
36 views

Equivalence of Galois groups of two different splitting fields of the same polynomial

All fields are in $\mathbb{C}$ Let $f$ be a polynomial with coefficients in the field $F$. Let $F_1$ be a Galois extension of $F$ such that its Galois group $G(F_1/F)$ is cyclic and has prime order. ...
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81 views

Coverings and elliptic $\mathbb{Q}$-curves

Following http://mathoverflow.net/questions/149815/automorphisms-of-the-l-function-associated-to-an-elliptic-mathbbq-curve I consider a $Q$-curve $E/K$ defined over $K$. If I'm not mistaken, the ...
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0answers
34 views

Shortened Reed-Solomon proving p(D) is primitive

Assume we have a shortened $(n=18, k=12, t=3)$ Reed Solomon code in $GF(2^{8})$.Let $\alpha$ be a primitive element of $GF(2^{8})$. Consider the primitive polynomial given by: $p(D) = D^{8} + D^{4} + ...
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votes
0answers
48 views

Bourbaki's proof of normal basis theorem Part 1

I have a few questions about Bourbaki's proof of normal basis theorem on Galois theory. This is part one. Let $K/k$ be a finite Galois extension, $G$ its Galois group. Let $k[G]$ be the group ring ...
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vote
1answer
45 views

Galois group of $x^5-x+1$ over $\mathbb{F}_7$

Let $K$ be the splitting field of $f(x)=x^5-x+1=(x^2+x+3)(x^3-x^2-2x-2)$ over $F=\mathbb{F}_7$. I want to find $\text{Gal}(K/F)$. Let $\alpha_1,\alpha_2$ be the roots of $x^2+x+3$ and ...