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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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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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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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 ...
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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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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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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 ...
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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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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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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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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 ...
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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 ...
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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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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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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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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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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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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 ...
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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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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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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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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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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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$H$-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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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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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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$\mathbb{Q}(\sqrt{2+\sqrt{2}})$ 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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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
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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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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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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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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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Determine with proof the number of monic irreducible polynomials of prime degree $p$ over $F$

Let $F$ be a field with $|F|=q$. Determine with proof the number of monic irreducible polynomials of prime degree $p$ over $F$, where $p$ need not be the characteristic of $F$. I know that ...
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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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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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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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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 ...
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Inverting the elements of a basis of a finite field extension

Let $K/k$ be a finite field extension of degree $d$. Suppose that $\{a_1, \dots, a_d\}$ is a basis of $K$ as a $k$-vector space. Is it true that $\{a_1^{-1}, \dots, a_d^{-1}\}$ is a basis of $K$ as a ...
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Why study solvability of polynomials of prime degree?

Galois investigated the solvability by radicals of polynomials of prime and prime-squared degree. These results are presented in modern form in chapter 15 of Cox's book on Galois Theory. Here is an ...
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How is the degree of the minimal polynomial related to the degree of a field extension?

I was reading through some field theory, and was wondering whether the minimal polynomial of a general element in a field extension L/K has degree less than or equal to the degree of the field ...
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The field of rational functions of $k$ contains an algebraic closure of $k$?

If $k$ is a field and $k(\mathbf{x}) = k(x_1,\dots,x_m)$ is the field of rational functions in the indeterminates $x_1,\dots,x_m$, then if $f(y) \in k(\mathbf{x})[y]$ is irreducible, take the ...
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Why differential Galois theory is not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation. My ...
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If K is a perfect field, is any extension of K also a Galois Extension?

So I have got that if K is a perfect field then every irreducible polynomial over K is separable. (Separable meaning it has no repeated roots). Is it then possible to conclude that every finite ...
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Why does the automorphism mapping $\omega$ to 1, not an element of Galois group

Let $L/\mathbb{Q}$ be a field extension, where $L=\mathbb{Q}(\omega)$ and $\omega=e^{\frac{2\pi i}{7}}$. In my textbook it states that $Aut(L/\mathbb{Q})=\{\sigma_i|\sigma_i(\omega)=\omega^i, 1\leq ...
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Prove that $x^p - t $ is irreducible in $\mathbb{F}_p(t)[x]$.

Prove that the polynomial $x^p - t$ is irreducible in $\mathbb{F}_p(t)[x]$. (Here $t$ is a formal variable). I know how to prove by Eisenstein's (for integral domains and ideals). However, my ...
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About the construction of resolvents in Galois theory (over $\mathbb{Q}$ in $\mathbb{C}$)

I have to say that my question is quite long and I apologize for this. The main idea is that I would like to show how to construct resolvents for any transitive subgroup of the permutation group to ...
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Why are separable and normal field extensions so called?

To my understanding: A separable extension $K/F$ is one in which the minimal polynomial of every $\alpha\in K$ has no multiple roots. A normal extension $K/F$ is one in which some polynomial $f\in ...
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Definition of Solvable by Radicals

I am currently trying to understand the definition of solvability by radicals. In the definition he takes a polynomial $f(x)$ over the field $K$ and then considers its splitting field $L$ and requires ...
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Explicitly construct a field with 729 elements

I would like to construct a field with 729 elements. I know that $729 = 3^6$, and that I have to find an irreducible monic polynomial of degree 6 over $GF(3)$. I chose the polynomial $x^6 + 2x^2 + 1$, ...
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1answer
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Simplifying the Splitting field of $x^n-a$

Let $L/K$ be a field extension where $L$ is the splitting field of the polynomial $x^n-a\in K[x]$. Clearly $L=K(t,\zeta t,\ldots,\zeta^{n-1}t)$, where $t=\sqrt[n]{a}$ and $\zeta$ is the primitive ...
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If $G$ and $N$ are soluble, then $G/N$ is soluble

Let $N$ be a normal subgroup of $G$. If both $G$ and $N$ are soluble, then $G/N$ is also soluble. My attempt: $G$ is soluble, so there exists a subnormal series $1=G_0\subset ... \subset G_n=G$. I ...