# Thread: Easy Galois Theory Questions

1. ## Easy Galois Theory Questions

Using permutation group notation:

Question 1: Is e, (12)(34), (13)(24), (14)(23) a representation of C2 x C2? [Note C2 = cyclic group of order 2.]

Question 2: Is C2 x C2 the group of symmetries of the following 4 numbers (viewed as roots of a polynomial with coefficients in Z)?
1-sqrt(2)
1+sqrt(2)
1-sqrt(3)
1+sqrt(3)

Question 3: If the above 2 assertions are correct, then can I conclude that C2 x C2 is the Galois Group of the polynomial:
(x^2 - 2x - 1)(x^2 - 2x - 2) = (x^4 - 4x^3 +x^2 + 6x + 2)?

(since the first quadratic has (as roots) the first 2 numbers listed, and the 2nd

2. ## Re: Easy Galois Theory Questions

Originally Posted by qmech
Using permutation group notation:

Question 1: Is e, (12)(34), (13)(24), (14)(23) a representation of C2 x C2? [Note C2 = cyclic group of order 2.]

Question 2: Is C2 x C2 the group of symmetries of the following 4 numbers (viewed as roots of a polynomial with coefficients in Z)?
1-sqrt(2)
1+sqrt(2)
1-sqrt(3)
1+sqrt(3)

Question 3: If the above 2 assertions are correct, then can I conclude that C2 x C2 is the Galois Group of the polynomial:
(x^2 - 2x - 1)(x^2 - 2x - 2) = (x^4 - 4x^3 +x^2 + 6x + 2)?

(since the first quadratic has (as roots) the first 2 numbers listed, and the 2nd
Why don't we work through this together? So, for Question 1 I'll assume the set of four elements of $\displaystyle S_4$ form a subgroup, call it $\displaystyle H$. We know from basic group theory that any group of order $\displaystyle p^2$ for prime $\displaystyle p$ is abelian, and so $\displaystyle H$ is abelian. But, by first principles (or the FTFGAG) we may conclude that $\displaystyle H\cong \mathbb{Z}_4,\mathbb{Z}_2^2$, clearly the deciding factor is whether there is an element of order $\displaystyle 4$ or not. Is there?