# Questions tagged [galois-theory]

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 (galois-connections).

6,682 questions
Filter by
Sorted by
Tagged with
13 views

### Proof that any extension that is Galois in the classical sense is also Hopf Galois

The usual action of $G$ on $L$ is by automorphisms that fix $\cal{K}$. Explicitly, for any $g\in G, l,m\in L, k\in \cal{K}$ \begin{align*} g(l+m)&=g(l)+g(l)+g(m)\\ g(lm)&=g(l)g(n)\\ g(k)&=...
31 views

### Extensions of an algebraically closed field

First some definitions: A field $k$ is algebraically closed if every non-constant polynomial $f(x) \in k[x]$ has a zero in $k$.Then if I am adding another restriction that if the field is also perfect,...
38 views

### Understanding why some Galois extension over $\mathbb{Q}_2$ has Galois group $\operatorname{GL}_2(\mathbb{F}_3)$

Let $K = \mathbb{Q}_2$ and $d_2 = 3x^4 + 12 x^2 + 4x - 4 \in K[x]$. Let $L_0$ be the splitting field of $d_2$ and let $\alpha \in L_0$ be a root of $d_2$. Let $L/L_0$ be the quadratic extension ...
10 views

### $E$ is the splitting field over $F$ for some separable $f(X) \in F[X]$ implies that $f(X)$ is separable over $\Phi(\Gamma(E/F))$?

I'm going over the proof of Theorem 25.1 in Abstract Algebra, and I'm trying to understand a particular detail concerning separable polynomials. Here's the relevant portion of the theorem statement: ...
13 views

38 views

### Extension of a field monomorphism to an automorphism .

$\mathbf {The \ Problem \ is}:$ Let $E$ be the splitting field of a polynomial $f$ over $k.$ Let $k \subset K \subset E,$ then show that any $k-$monomorphism $\phi$ from $K \to E$ can be extended to a ...
55 views

### E/F is Galois ext with $[E:F]=p^2$ and its Galois group is not cyclic, then $\exists K$ a proper subfield of $E$.

Let $E$ be a Galois extension of a field $F$. Suppose that the Galois group $Gal(E/F)$ is an abelian group of order $p^2$ which is not cyclic. Then I want to show $\exists K$ a proper subfield of $E$...
30 views

39 views

### Proving the irreducibility of a polynomial based on its Galois group

Suppose $f(X) \in \mathbb{Q}[X]$ is a polynomial of degree $n$. Let $K$ be the splitting field of $f(X)$ over $\mathbb{Q}$. Prove that if $Gal(K/\mathbb{Q}) \simeq S_n$, then $f(X)$ is irreducible ...
29 views

51 views

86 views

### Existence of a Field not containing $\sqrt{2}$, such that any finite extension is cyclic.

I want to show the existence of a field $E$ not containing $\sqrt{2}$, such that any finite extension of $E$ in $\overline{\mathbb{Q}}$ is cyclic. I think the maximal field not containing $\sqrt{2}$ ...
26 views

### An isomorphism related to the class group

Suppose that $L/K$ is a Galois extension with abelian Galois group. Let $\phi : Cl(\mathcal{O}_K) \rightarrow Cl(\mathcal{O}_L)$ be the morphism of class groups given by $\phi(I) = I\mathcal{O}_L.$ My ...
53 views

### What does 'topologically' generated mean?

I heard ablolute galois group of finite field is 'topologically' generated by frobenius map. I understand this sentence except for 'topologically', but what does 'topologically' exactly mean here? ...
39 views

### Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$.

I found the following problem: Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$. Can someone give me a hint? I remember ...
18 views

38 views

### Why is the extension $L/F$ in the proof is normal$?$

In proof of part $(1)$ of this preposition, $L/F$ is a galois extension. However, I didn't understand why. It is a separable extension. But how can one prove the normality$?$ Here, $E/F$ is a finite ...
46 views

### Is $x$ always a primitive element of $\text{GF}(2^m)$?

I checked using MATLAB that $x$ is a primitive element of $\text{GF}(2^m)$ for $m\le 16$. Is the statement true for $m > 16$? EDIT: In this question, we represent an element of $\text{GF}(2^m)$ as ...
39 views

### Elements of Galois group map roots of minimal polynomial to which other roots?

If we have $F \leq E \leq \mathbb{C}$, I know that for any $\alpha \in E$ and $\sigma \in \operatorname{Gal}(E/F)$, we have that $\sigma$ maps roots of $m_{\alpha,F}(x)$ to roots of $m_{\alpha,F}(x)$. ...
### How to compute the Galois Group of $(x^3-3)(x^4-2)$ over $\mathbb Q$
So I know the splitting field is $\mathbb Q(\sqrt2, i ,\sqrt3,\zeta_3 )$ where $\zeta_3=-\frac12+\frac{\sqrt3}{2}i$ is the root of unity. And the order of the Galois Group equals the degree of ...
### Galois group of $\mathbb{Q}_p(\zeta_4,\sqrt{p})/\mathbb{Q}_p$ for $p-1$ not divisible by $4$
Let $p > 2$ be a prime number such that $p-1$ is not divisible by $4$. Consider $K = \mathbb{Q}_p$ and $L = \mathbb{Q}(\sqrt{p},\zeta_4)$ where $\zeta_4 \in L$ is a primitive forth root of unity ...