# Questions tagged [galois-extensions]

For questions about Galois extensions fields. We say $L/K$ is a Galois extension iff the subfield of $L$ that is fixed by automorphisms of $L$ which fix K is exactly $K$.

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### What is the Galois group of the polynomial $f(x)=x^3-3$ over $\mathbb{Q}$?

What is the Galois group of the polynomial $f(x)=x^3-3$ over $\mathbb{Q}$ ? $f(x)=0 \$ gives $x= 3^{1/3},~ \zeta_3 3^{1/3},~ \zeta_3^2 3^{1/3}$ over $\mathbb{C}$. Thus the permutation of these $3$ ...
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### What is the intuition behind mapping of elements from $GF(2^8)$ to $GF(((2^2)^2)^2)$?

I'm finding it very difficult to understand the concept of mapping elements from the extension field $GF(2^8)$, to $(GF(2)^2)^2)^2$. I realize that the field that the elements of the field, $GF(2^8)$,...
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### Abelian extensions $E|F$ and $F|K$ $\implies E|K$? [closed]

Let $E,F,K$ fields such that $K\subset F \subset E$ If $F|K$ and $E|F$ are abelian extensions, then $E|K$ is an abelian extension? I can't find a counter example
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### Proof of : If $K$ is a finite extension of $F$, then $o(G(K,F)) \leq [K:F]$

$G(K,F)$ is the group of all automorphisms of $K$ that leave $F$ fixed. $F$ is a field of characteristic zero. $[K:F]$ is the dimension of $K$ as a vector space over $F$. Herstein proves this theorem (...
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### Criterion for completely splitting primes in a radical extension

Given $d \in \mathbb{Z}^+$ and $a \in \mathbb{Q}^*$, let $K = \mathbb{Q}(\zeta_d, a^{1/d})$, where $\zeta_d$ is a primitive $d$th root of unity. I'm trying to prove that: "A prime number $p$ splits ...
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### Let $X = \{ \sqrt{p} : p \text{ is prime} \}$, $Y \subseteq X$ and $\sqrt{p} \not\in Y$. Show that $[\mathbb{Q}(Y)(\sqrt{p}) : \mathbb{Q}(Y)] = 2$.

I am trying to solve Problem 22 from Chapter 5 of Patrick Morandi's Field and Galois Theory: Let $K = \mathbb{Q}(X)$, where $X = \{ \sqrt{p} : p \text{ is prime} \}$. Show that $K$ is Galois ...
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### Splitting of primes and other properties of $\mathbb{Q}[\omega]$ for $\omega=e^{2\pi i/m}$

Reading through Marcus I came to this exercise part of which already have answers in this same site (Splitting of primes in real cyclotomic field ) but no complete answer can be found and I'm having ...
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### Description of the decomposition and the inertia group in terms of the product $\mathbb{Z}^*_{p^k}\times \mathbb{Z}^*_n$

Let $\omega^{\frac{2\pi}{m}}$, we fix a prime p and write $m=p^kn$ with $p\not| \, n$. We know that the Galois group of $\mathbb{Q}[\omega]$ over $\mathbb{Q}$ is isomorphic to $\mathbb{Z}^*_m$ that ...
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### Find Galois group of $\mathbb{Q}\sqrt{(2+\sqrt2)(3+\sqrt3)}$ [duplicate]

Find Galois group of $Q=\mathbb{Q}\sqrt{(2+\sqrt2)(3+\sqrt3)}$. I know the minimal polynomial is $f(x)=x^8 -24x^6+144x^4-288x^2+144$. It's irreducible over $\mathbb{Q}$, hence, $[Q:\mathbb Q]=8$. ...
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### Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$

Find Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$ and describe an extension corresponding to any of it's proper subgroups of maximal order. I know that the roots are \sqrt{\frac{1}{3}\left(2 ...
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### Intuition for equivalent definitions of Galois extension

Assume $K/F$ is a finite field extension. Then the following are equivalent: $|\operatorname{Gal}(K/F)| = [K:F]$. The fixed field of $\operatorname{Gal}(K/F)$ is $F$. $K$ is the splitting field of ...
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### Is $\operatorname{Gal}(\overline{\mathbb{F}_p} \,/\, \mathbb{F}_p)$ cyclic?

I know if $L/K$ is a extension of finite fields then $\text{Gal}(L/K)=\langle \phi \rangle$ where $\phi:L\longrightarrow L$ is the Frobenius automorphism. How can I show that it is also true for the ...
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### Degree of extension $\mathbb{C}/K$, where $K$ is maximal with the property $\sqrt{2} \notin K$

This question has been asked before but not really answered, but my query is a bit separate. To summarise the details: $K$ is a field maximal with respect to the property $\sqrt{2}\notin K$, any ...
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### Divide any given angle into $n$ equal parts using paper folding

I have known that it is possible to trisect an arbitrary angle by Hisashi Abe's method. Of course, it is easy to divide an angle into two or four equal parts using paper folding. My question is ...
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### Galois Group of $x^{6}-2x^{3}-1$

I was trying to compute the normal closure of $\mathbb{Q}[\alpha]$, where $\alpha = \sqrt{1+\sqrt{2}}$. I had a reallyyyy hard time proving that $x^{6}-2x^{3}-1$ is irreducible. I proved that it ...
### Arithmetic in GF$(2^{32})$ using GF$(2^{16})$ and extensions
Ultimately, I'm looking to implement arithmetic in GF$(2^{32})$. I have a library that implements arithmetic in GF$(2^{16})$ using look-up tables for log and anti-log to implement multiplication, and ...