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Trouble with two Galois theory questions

LARaiders

New member
May 8, 2013
1
I am so confused with these two questions. Can anyone help me out?

1) Please find [Q((√7 , √5) : Q] by finding f(x) such that Q (√7 , √5) ≅ Q[x]/(f(x)),

2) Prove that φ(4root√3) = ± 4root√3, Given that φ ∈ Gal(Q(4root√3)|Q)
 

Siron

Active member
Jan 28, 2012
150
1. What you have to do is to find the minimal polynomial of [tex]\sqrt{7}[/tex] and [tex]\sqrt{5}[/tex], that wil give you the degree [tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}][/tex].
Remark, note that
[tex][\mathbb{Q}(\sqrt{5},\sqrt{7}):\mathbb{Q}]=[\mathbb{Q}(\sqrt{5},\sqrt{7}),\mathbb{Q}(\sqrt{7})][\mathbb{Q}(\sqrt{7}):\mathbb{Q}][/tex]
 

TheBigBadBen

Active member
May 12, 2013
84
1)
Try

\(\displaystyle f(x) = (x^2-7)(x^2-5)\)

2)

What does

\(\displaystyle \phi \in Gal(\mathbb{Q}(4 \sqrt{3})|\mathbb{Q})\) do to the roots of the minimal polynomial?