Articles of galois theory

Is every subgroup of automorphisms a Galois group?

Let $G=\{\sigma_1=1, \sigma_2, \dots,\sigma_n\}$ be a subgroup of automorphisms of a field $K$ and let $F$ be the fixed field. Then $[K:F]=|G|$ Why is this always true? I thought this would be true only if $K$ was the Galois extension of $F$ and $G$ was $\text{Aut}(K/F)$

The field of Laurent series over $\mathbb{C}$ is quasi-finite

How can I prove that the field of Laurent series over $\mathbb{C}$ is quasi-finite, which means that it has a unique extension of degree $n$ for all $n \geqslant 1$ ? The article says that the extension of degree $n$ is $\mathbb{C}((T^{1/n}))$. I think that I understand why it is an extension of degree […]

Applying Extended Euclidean Algorithm for Galois Field to Find Multiplicative Inverse

I was trying to apply the Extended Euclidean Algorithm for Galois Field. Among the many resources available, I found the methodology outlined in this document easy to grasp. The above works fine when applied to numbers. Now, for $\textit{GF}(2^3)$, if I take the polynomial $x^2$ and the irreducible polynomial $P(x) = x^3 + x + […]

Finding Galois group of $x^6 – 3x^3 + 2$

I’m trying to find the Galois group of $$f(x)= x^6 – 3x^3 + 2$$ over $\mathbb{Q}$. Now I can factorise this as $$f(x) = (x-1)(x^2 + x + 1)(x^3 – 2)$$ I can see the splitting field must be $\mathbb{Q}(\omega, \sqrt[3]{2})$ (where $\omega$ is a 3rd root of unity) which has degree $6$ and so […]

Finding the size of a Galois Group of a splitting field for a polynomial of degree 6.

Let $f(x) = x^6 + ax^4 + bx^2 + c$ with a,b,c ∈ $\mathbb{Q}$ be an irreducible polynomial in $\mathbb{Q}$[x]. Let K be the splitting field of f(x) over $\mathbb{Q}$ and let G = Gal[K:$\mathbb{Q}$]. Then prove that |G| ≤ 48. My attempt: I have that [$\mathbb{Q}$(α) : $\mathbb{Q}$] = 6. I claim that one […]

The product map and the inverse map are continuous with respect to the Krull topology

Let $K/F$ be a Galois extension with $G=\text{Gal}(K/F)$. $\mathcal{I}=\lbrace E\text{ }|\text{ }E/F \text{ is Galois and }[E:F]<\infty\rbrace$ $\mathcal{N}=\lbrace N\text{ }|\text{ } N=\text{Gal}(K/E) \text{ for some } E\in\mathcal{I}\rbrace$ The Krull topology over $G$ is defined as follows. The set $B=\lbrace \sigma_iN_i\text{ }|\text{ }\sigma_i\in G \text{ and }N_i\in\mathcal{N}\rbrace$ is the base for the Krull topology. I am […]

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 that L/F is a normal extension and if we have $ […]

Cyclotomic polynomials and Galois groups

According to this question I want to extend the question from there. Lets consider again the galois extension $\mathbb Q(\zeta)/\mathbb Q$ where $\zeta$ is a primitive root of the $7^{th}$ cyclotomic polynomial. I want to determine the minimal polynomial of $\zeta+\zeta^{-1}$ and $\zeta+\zeta^{2}+\zeta^{-3}$. I know that one of the minimal polynomial has degree 2 and […]

Minimal cyclotomic field containing a given quadratic field?

There was an exercise labeled difficile (English: difficult) in the material without solution: Suppose $d\in\mathbb Z\backslash\{0,1\}$ without square factors, and $n$ is the smallest natural number $n$ such that $\sqrt d\in\mathbb Q(\zeta_n)$, where $\zeta_n=\exp(2i\pi/n)$. Show that $n=\lvert d\rvert$ if $d\equiv1\pmod4$ and $n=4\lvert d\rvert$ if $d\not\equiv1\pmod4$. It’s easier to show that $\sqrt d\in\mathbb Q(\zeta_n)$, although I […]

Show $F(U) = K((x^q -x)^{q-1})$.

Let $K$ be a finite field with $q$ elements. Show that if $U$ is the subgroup of $Aut(K(x)/K)$ which consists of all mappings $\sigma$ of the form $(\sigma \theta)(x) = \theta(ax+b)$ with $a \neq 0$ then $F(U) = K((x^q -x)^{q-1})$. I am not getting any clue to solve the problem. Help Needed. Here $F(U)$ is […]