Determine the Galois groups of the following polynomials over the fields indicated:
(i) $x^{4}-4x^{2}+5$ over $\mathbb{Q}$
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4What did you try? – José Carlos Santos Mar 08 '18 at 17:55
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3Find the roots first. Use the quadratic formula. – Dionel Jaime Mar 08 '18 at 17:58
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So, I got that the resolvant cubic is $x^3+4x^2-20x-80$, which has roots $-4, -2\sqrt{5}, 2\sqrt{5}$. But I don't know what $[K(-4,-2\sqrt{5},2\sqrt{5}) : K]$ is. – Tim Mar 08 '18 at 18:24
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What does $ K $ denote? – Bernard Mar 08 '18 at 18:28
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$K$ is a field and $f \in K[x]$ – Tim Mar 08 '18 at 18:32
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Ok, just got that because it's over $\mathbb{Q}$, I'm supposed to be finding $[\mathbb{Q}(-4,-2\sqrt{5},2\sqrt{5}):\mathbb{Q}]$ – Tim Mar 08 '18 at 18:48
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2@Tim That's $|\Bbb Q(\sqrt5):\Bbb Q|$. – Angina Seng Mar 08 '18 at 19:09
1 Answers
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This can be decided by the discriminant $disc(f)$ and the cubic resolvent $R_3(f)$ of $f$, see K. Conrad's notes Galois groups of cubics and quartics, Theorem $3.6$ and following. Corollary $4.5$ shows how to find it for all irreducible quartics of the form $$ f(x)=x^4+bx^2+d. $$ The possibilities are either $V_4,D_4$ or $C_4$.
Dietrich Burde
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