Ganesh Ujwal
New member
- Joined
- Aug 10, 2014
- Messages
- 32
Identifying the Galois Group G(Q[2,32]/Q)
I am trying to determine the Galois group G(Q[2,32]/Q). I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just taught myself this material today.
First, note that K=Q[2,32] is a field extension of degree 6. However, it is not a splitting field of some polynomial with coefficients in Q. Thus, the order of the Galois group will be strictly less than $6$ since it is not a Galois extension.
Consider the polynomial f(x)=(x2−2)(x3−2). This polynomial has 3 roots in K, namely 2, −2, and 32. Any Q-automorphism of K will permute these three roots. Note that for any such automorphism, 0=ϕ(0)=ϕ(2−2)=ϕ(2)+ϕ(−2). Hence, the only two possible automorphisms are the identity and that which swaps 2 with −2.
We conclude that G(K/Q)≅Z2.
I am trying to determine the Galois group G(Q[2,32]/Q). I am fairly confident I have the correct answer, but I need someone to confirm my work since I have just taught myself this material today.
First, note that K=Q[2,32] is a field extension of degree 6. However, it is not a splitting field of some polynomial with coefficients in Q. Thus, the order of the Galois group will be strictly less than $6$ since it is not a Galois extension.
Consider the polynomial f(x)=(x2−2)(x3−2). This polynomial has 3 roots in K, namely 2, −2, and 32. Any Q-automorphism of K will permute these three roots. Note that for any such automorphism, 0=ϕ(0)=ϕ(2−2)=ϕ(2)+ϕ(−2). Hence, the only two possible automorphisms are the identity and that which swaps 2 with −2.
We conclude that G(K/Q)≅Z2.