Linked Questions

5
votes
1answer
2k views

Computing the Galois group of $x^4+ax^2+b \in \mathbb{Q}[x] $ [duplicate]

I want to compute the Galois group of some polynomials, but I want to see some examples first. For example this proposition could be helpful. I don't know how to prove it <.< Let's consider a ...
0
votes
1answer
50 views

Galois group of $x^4 - 2x^2 - 6$ - generators [duplicate]

The splitting field of $x^4 - 2x^2 - 6$ is $\mathbb{Q}(\alpha, \sqrt{-6}) = \mathbb{Q}(\alpha, \beta)$ where $\alpha = \sqrt{1+\sqrt{7}}$, $\beta = \sqrt{1-\sqrt{7}}$. From the first representation, $...
0
votes
0answers
31 views

How to determine the Galois group of $x^4-x^2-1$ over $\mathbb{Q}$ [duplicate]

Considering $L$ as the splitting field of $x^4-x^2-1$ over $\mathbb{Q}$, determine the group Gal$(L/\mathbb{Q})$.
5
votes
2answers
186 views

Prove that $[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$.

I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ...
6
votes
1answer
462 views

Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$

Suppose $L=\mathbb{Q}(\sqrt{a+b\sqrt{d}})$,($d$ and $a+b\sqrt{d}$ are square free algebraic integers) when is $L/\mathbb{Q}$ a normal extension? When does $Aut(L/\mathbb{Q})=\mathbb{Z}/4\mathbb{Z}$ or ...
4
votes
1answer
177 views

The Galois group is $\mathbb{Z_{4}}$ if and only if $\frac{\alpha}{\beta}-\frac{\beta}{\alpha}\in\mathbb{Q}$

This question is from Lang's Algebra Chapter VI Exercise Q8 Let $f(x)=x^4+ax^2+b$ be an irreducible polynomial over $\mathbb{Q}$, with roots $\pm\alpha$, $\pm\beta$ and splitting field $K$. I have ...
2
votes
2answers
117 views

cyclic extension of prime power of a local field

Let $K$ be a non archimedian local field of characteristic $p>0$ with residue field $\mathbb{F}_p$ and $l\neq p$ be a prime. It is known by local classfieldtheory that any abelian Galois ...
4
votes
2answers
80 views

$\alpha \alpha' \in K^2 $ if and only if $Gal(E/K) \cong C_2 \times C_2$

K= field of characteristic p $\neq$ 2, $c \in K-K^2$ and $F=K(\sqrt{c})$. Let $\alpha = a+b\sqrt{c}$ such that $\alpha \notin F^2$ and $E=F(\sqrt\alpha)$. Define $\alpha'=a-b\sqrt{c}$. How can I ...
1
vote
2answers
89 views

Find the Galois group of $\mathbb{Q}(\sqrt{4+\sqrt{7}})/\mathbb{Q}$ [duplicate]

Let $E = \mathbb{Q}(\sqrt{4+\sqrt{7}})$. Prove that $E$ is a normal extension of $\mathbb{Q}$ and find the Galois group $Gal(E/\mathbb{Q})$. My attempt: Let $\alpha = \sqrt{4+\sqrt{7}}$. $\alpha$ is ...
0
votes
0answers
113 views

Classify the Galois group of tower of degree $2$ extension.

For a field $F$ such that $char(F)\ne 2$. Consider the tower of extension $E/L/F$ where $L = F(√c)$ and $E = L(\sqrt{a + b\sqrt{c}})$, for some $a,b,c ∈ F$. When the degree-$4$ extension is Galois, ...
1
vote
2answers
73 views

When is the extension $\Bbb{Q}(\sqrt{\sqrt{5} +a})/\Bbb{Q}$ Normal?

For what values of $a \in \Bbb{Q}$ is the extension $\Bbb{Q}(\sqrt{\sqrt{5} +a})/\Bbb{Q}$ Normal? I know that a finite extension $L/K$ such as this is normal iff it is the splitting field of some $f \...
2
votes
0answers
90 views

Properties of Quadratic Field Extensions

Let $F\subset L\subset E $ be a tower of Field Extensions such that $F$ does not have characteristic $2$ and $[E:L]=[L:F]=2$. We know that $L=F(\sqrt{c})$ and $E=L(\sqrt{a+b\sqrt{c}})$ for some $a,b,c\...