# 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 $K$.

541 questions
Filter by
Sorted by
Tagged with
67 views

### Computing a certain Galois group

I am trying to answer this question: Suppose $p$ is an odd prime and $K/\mathbb{Q}$ is the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of 1 in $\mathbb{C}$. (a) Show that $K$...
31 views

13 views

### If $f(t,x)\in F_q(t)[x]$ is a Morse function, does this mean splitting field of $f(t,x)$ over $F_q(t)$ is a regular extension?

One of the classical result of Hilbert says, if $f$ is a Morse function, then the splitting field of $f(X,T)$ over $Q(T)$ is a regular extension with Galois group $S_n.$ J. P.Serre- Topics in Galois ...
58 views

### Why doesn't “$S_n$ appears as a Galois group over $\Bbb{Q}$” wrap up the Inverse Galois Problem?

Since every finite group $G$ is embedded in $S_n$ for $n = |G|$ and Hilbert showed that $S_n$ appears as a Galois group of $K/\Bbb{Q}$ for some Galois extension $K$, then how does that not wrap up the ...
40 views

### $\mathbb{Q}(\alpha)$ extension of degree 3 is galois over $\mathbb{Q}$ if and only if discriminant of minimal polynomial of $\alpha$ is square.

I supposed that $\alpha$, $\beta$ and $\gamma$ are the roots of the minimal polynomial in its splitting field. So the discriminant is $(\alpha-\beta)^2(\alpha-\gamma)^2(\beta-\gamma)^2$. If every ...
19 views

### Hopf-Galois structures of cyclic type on a dihedral or quaternionic extension

Let $L/K$ be a dihedral or quaternionic finite field extension, that is such that $Gal(L/K)$ is either a dihedral or a quaternion group. How many Hopf-Galois structures of cyclic type are there on ...
63 views

### $K$ be a finite extension of $\mathbb Q$ then there exists a finite extension $E$ over $K$ such that $E/K$ is not normal.

Let $K$ be a finite extension of $\mathbb Q$. Then I have to show that there exists a finite extension $E$ over $K$ such that $E/K$ is not normal. It is clear that $E/\mathbb Q$ is not normal as well....
51 views

### Adding finite list of square roots of primes to $\mathbb{Q}$

I have that idea which I'm pretty sure that is true- but I haven't succeded to prove it: Given finite list of primes ${p_1,p_2,p_3,...,p_n}$ , and extension of the rational field with the square roots ...
111 views

### Finite Galois group of Galois extension implies that the extension is finite?

Assume that the field extension $K \subset L$ is a Galois (in other words: normal and separable, possibly infinite) extension with finite Galois group. If one starts with a finite Galois extension, ...
35 views

### Is the extension of a perfect field by one of its algebraic closures always a Galois extension?

Let $\overline{K}$ be an algebraic closure of a perfect field $K$. I have read that the absolute Galois group of a perfect field is simply $\text{Gal}(\overline{K}/K)$. Yet if Galois groups are only ...
79 views

### Is $\mathbb{Q}(\cos\frac{2\pi}{13})/\mathbb{Q}$ a Galois extension?

I've been solving problems from my Galois Theory course, and I got stuck in this problem: Calculate the number of subfields of $\mathbb Q(\cos\frac{2\pi}{13})$ What I considered doing is finding the ...
58 views

### Showing the extension $\mathbb{Q}(\sqrt{-3})/\mathbb{Q}$ is Galois and determining its Galois group.

I'm trying to show that $\mathbb{Q}(\sqrt{-3})/\mathbb{Q}$ is a Galois extension. I would like to show that $\mathbb{Q}(\sqrt{-3})/\mathbb{Q}$ is the splitting field of $f(x) = x^6 - 3$ which is ...
71 views

### What is $\text{Frob}\in \text{Gal}(\overline{K}/K)$?

Let $\overline{\rho}: \text{Gal}(\overline{K}/K)\to \text{GL}_2(\mathbb{F}_q)$ be an unramified Galois representation at a place $v$ of $K$. Since $\ker(\overline{\rho})$ is closed, it corresponds to ...
27 views

40 views

### Prove $\operatorname{Gal}(M_3/M_1)\cong\operatorname{Gal}(M_2/(M_1\cap M_2))$

I've been solving problems from my Galois Theory course, and I got the idea to solve this one but I'm unable to prove some details. The problem goes like this: Being $L/K$ a Galois extension, and ...
18 views

### Normal closure as the compositum of conjugates

An excerpt from Serge Lang's Algebra Chapter V $\S4$ p. 242. Let $E$ be a finite extension of $k$. The intersection of all normal extensions $K$ of $k$ (in an algebraic closure $E^\text{a}$) ...
46 views

87 views

66 views

### Textbook's proof of Galois correspondence is circular

I am reading Galois Theory 4th edition by Ian Stewart and the proof of one part of the Galois correspondence is circular. $L : K$ is a finite normal field extension inside $\mathbb{C}$ with Galois ...
75 views

### Question regarding galois extension and galois groups

From what I have read, for a field extension $E/F$, the automorphism group $Aut(E/F)$ is a Galois group when $E/F$ is Galois, that is, the extension is normal and separable, or equivalently, when $E$ ...
Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For ...
### Is the quotient field of $\mathbb{Z}[\sqrt{-3}]$ galois over $\mathbb{Q}$ and what is the Galois group?
I proved that $\mathbb{Z}[\sqrt{-3}]$ is isomorphic to $\mathbb{Z}[x]/(x^{2}+3)$. Because $f = (x^{2}+3)$ is irreducible by Eisenstein. Is that enough to conclude that it splits over $\mathbb{Q}$? If ...