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### 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 ...
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, $... 0answers 31 views ### How to determine the Galois group of$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})$. 2answers 186 views ### Prove that$[\mathbb{Q}(\sqrt{4+\sqrt{5}},\sqrt{4-\sqrt{5}}):\mathbb{Q}] = 8$. I'm trying to prove this result using elementary Field and Galois theory, but in an "efficient" way. It is desirable to avoid the use of powerful theorems of group theory or results about the ... 1answer 462 views ### Determine the Galois group of$\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}$or ... 1answer 177 views ### The Galois group is$\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$. I have ... 2answers 117 views ### cyclic extension of prime power of a local field Let$K$be a non archimedian local field of characteristic$p>0$with residue field$\mathbb{F}_p$and$l\neq p$be a prime. It is known by local classfieldtheory that any abelian Galois ... 2answers 80 views ###$\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}$. How can I ... 2answers 89 views ### Find the Galois group of$\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 ... 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, ... 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 \...
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\...