Question

Question: Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to Z₂.

Short answer

Answer:

Irreducible quadratic polynomial is isomorphic Z₂.

Step-by-step solution

Defintion of Galois extension.

If K is a finite dimension normal separable extension field of the field F then K is a Galois extension of F. In other word, K is Galois over F.

Step-2: Showing that irreducible quadratic polynomial is isomorphic Z2 .

Consider the irreducible quadratic polynomial$x^2-a$ of order 2. Let F be a splitting field then the chain of extension is

$Q\subset Q\left(\alpha \right)=F$

Where the root is the real root of the polynomial thus

$$\begin{gathered} \raisebox{1ex}{$F$}\!\left/ \!\raisebox{-1ex}{$Q=Q\left(\alpha \right)/Q$}\right. \\ =3 \end{gathered}$$

The Galois group is the subgroup of S₂ of order 2. Since group G is a subgroup of S₂ and all subgroups are conjugate hence isomorphic. Since group G is isomorphic to Z₂. Hence the Galois group is a subgroup of S₂ consist the irreducible quadratic polynomial which is isomorphic of order 2 to Z_2.