Question

Let$\mathbf{p}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathbf{and}\mathbf{}\mathbf{q}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ be irreducible quadratics. Prove that the Galois Group of $\mathbf{f}\left(x\right)\mathbf{=}\mathbf{q}\left(x\right)\mathbf{p}\left(x\right)$ is isomorphic to ${\mathbf{Z}}_{\mathbf{2}}\mathbf{\times }{\mathbf{Z}}_{\mathbf{2}}$ or ${\mathbf{Z}}_{\mathbf{2}}$.

Short answer

Answer:

The Galois group of $f\left(x\right)=q\left(x\right)p\left(x\right)$ is isomorphic to $Z_2\times Z_2$.

Step-by-step solution

Defintion of Galois extension

A splititing field of a minimal polynomial with coefficient is an extension field is a smallest field extension of that field over which the polynomial split into linear factors.

Step-2: Show that fx=qxpx  is isomorphic to  Z2×Z2 

Consider the splitting field$f\left(x\right)=q\left(x\right)p\left(x\right)$here the field is the product of two irreducible quadraic polynomial since

$$\begin{gathered} p\left(x\right)=x^2-a \\ q\left(x\right)=x^2-b \end{gathered}$$

And the splititing field is

$$\begin{gathered} f\left(x\right)=p\left(x\right)q\left(x\right) \\ =\left(x^2-a\right)\left(x^2-b\right) \\ =x^4-\left(a+b\right)x^2+ab \\ =x^4+c{x}^2+d \end{gathered}$$

Since $c=-\left(a+b\right)$ and d = ab if u is the root of p (x) and v is the root of q (x) there are the two possiblilities $v\notin F\left(u\right)$and $v\in F\left(u\right)$. Let $\alpha$ be the root of splititing field F (x) over F than $Q\left(\alpha ,\sqrt{b}\right)$ is contains all the root of . And Galois group of $Q\left(\alpha \right)$over the Q is the degree of 1,2 and 4 so, $Q\left(\alpha ,\sqrt{b}\right)$ over $Q\left(\alpha \right)$is a degree of 1 or 2. Since

$$\begin{gathered} Q\left(\alpha ,\sqrt{b}\right)/Q\left(\alpha ,\right)=\left(Q\left(\alpha ,/Q\right)\right)\left(Q\left(\alpha ,\sqrt{b}\right)/Q\left(\alpha \right)\right) \\ =1,2,4,8 \end{gathered}$$

There are the order of the splititing field. Hence the order of the Galos group is the same as the degree of the splititing field. If it is order 2 so the group is $Z_2\times Z_2$ and$Z_4$ if it is the order 4 so the group is or and the overall group is isomorphic to degree 4.