Question

Question: Prove that the Galois group of an irreducible quadratic polynomial is isomorphic toZ₂.

Short answer

Answer:

Irreducible quadratic polynomial is isomorphic Z₂.

Step-by-step solution

Defintion of Galois extension.

If Kis a finite dimension normal separable extension field of the fieldFso Kis a Galois extension of Fin other word Kis 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 splititing field then 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 subgroup 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 Z₂ .