# Math Help - 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
quadratic has the latter 2)

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

3. ## Re: Easy Galois Theory Questions

There is no element of order 4. All the permutations listed in Q1 are of order 2.
For instance: (12)(34)(12)(34) = e.
This multiplication (12)(34)(14)(23) = (13)(24) shows that these elements act like a group.

I'm pretty confident about Q1 because I can multiply things out explicitly. The heart of my question is Q2 and Q3.