Let $F\subset L\subset E $ be a tower of Field Extensions such that $F$ does not have characteristic $2$ and $[E:L]=[L:F]=2$. We know that $L=F(\sqrt{c})$ and $E=L(\sqrt{a+b\sqrt{c}})$ for some $a,b,c\in F$. When will $E/F$ be Galois?i.e., when can we say in terms of $a,b,c$ that $E/F$ is Galois, or, better, what can be said about the group $G(E/F)$? I think we have to use the quadratic formula here and the fact that an element $y\in A(\sqrt{x})$ is a square iff $y$ or $xy$ is a square in $A$, where $A$ is a field. Any ideas. Thanks beforehand.
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I think you may mean $E=L(\sqrt{a+b\sqrt{c}})$ – sharding4 May 23 '17 at 16:42
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@sharding4 edited the post. thanks – vidyarthi May 23 '17 at 16:43
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@user1952009 But $[E:F]=4$, hence, $E/F$ is not a quadratic extension, isnt it? – vidyarthi May 23 '17 at 16:58
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@user1952009 see here – vidyarthi May 23 '17 at 17:00
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Why not write the minimal polynomial of $\sqrt{a+b\sqrt{c}}$ and see what are its roots ? – reuns May 23 '17 at 17:10
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@user1952009 great idea. so then, $E/F$ is Galois when the quartic polynomial completes a square, isnt it? – vidyarthi May 23 '17 at 17:13
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1$\pm\sqrt{a+b\sqrt{c}}$ is a solution of $(x^2-a)^2 = b^2 c$ as well as $\pm\sqrt{a-b\sqrt{c}}$ so it is necessary and sufficient that $\sqrt{a-b\sqrt{c}} \in E$ which happens when $\sqrt{a^2 - b^2 c} \in L$, that's what you meant ? – reuns May 23 '17 at 17:18
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@user1952009 so, thats the answer! but, what can we say about the galois group $G(E/F)$? – vidyarthi May 23 '17 at 17:20
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For the classification of quartic Galois groups see there https://math.stackexchange.com/questions/649466/galois-group-of-a-quartic – reuns May 23 '17 at 17:21
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1This is exactly Serge Lang p.231 exercise 4. – Divide1918 Dec 30 '20 at 14:24