Question

Prove that the Galois group of an irreducible cubic polynomial is isomorphic to Z₃or W S₃.

Short answer

Answer:

Irreducible cubic polynomial is isomorphic .

Step-by-step solution

STEP 1:

If K is a finite dimension normal separable extension field of the field F so K is a Galois extension of F in other word K is Galois over F .

Step-2: Showing that irreducible cubic polynomial is isomorphic Z3

Consider the cubic polynomial $f\left(x\right)=a{x}^3+b{x}^2+cx+d$whose Galois group contain 3 cycles and a transponition. Let polynomial$f\left(x\right)=a{x}^3+b{x}^2+cx+d$ . Then F(X) is irreducible of modulo 3 thus the Galois group if F(X) is S₃ .

Consider the irreducible cubic polynomial$x^3-a$of order 3. Let F be a splititing field then chain of extension is

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

Here, the root is the real root of the polynomial thus

$$\begin{gathered} F/Q=Q\left(\alpha \right)/F \\ =3 \end{gathered}$$

The Galois group is the subgroup of S₃ of the 3. Since group G is a subgroup of S₃ and all subgroup are conjugate. Hence ismomorphic since group is isomorphic to Z₃. Hence the Galois group is a subgroup of consist the irreducible cubic polynomial which is isomorphic of order 3 to Z₃ . Hence the Galois group is isomorphic of S₃ or Z₃ .