Let M : L : K be finite field extensions. When M is not normal over K, give four examples to show that this gives no information about the normality of M over L or of L over K. What are the possibilities if M is normal over K?
Normality and Field Extensions
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abstract-algebra
field-theory
galois-theory
extension-field
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1Can you give just *any* example of a normal and a not normal extension? – 2012-11-06
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1@Hagen so $A=\mathbb{Q}(2^{1/2})$ is a splitting field for $a(x)=x^2 - 2$ because all roots of $a$ are in $A$ and since $a(x)\in\mathbb{Q}[x]$ I know that $A$ is a normal extension of $\mathbb{Q}$. Alternatively, let $B=\mathbb{Q}(2^{1/3})$ and $b(x)=x^3 - 2$ then not all roots of $b$ are in $B$ so $B$ is not a normal extension of $\mathbb{Q}$ – 2012-11-06
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0Now for the first part of the problem you might take $M=\mathbb Q(2^{1/3})$, $K=\mathbb Q$ and cheat a little by letting $L=M$ or $L=K$, respectively. For the second part, waht can you sa about $M:L$ if $M:K$ is normal? – 2012-11-06
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1Given that extensions of finite fields are always normal, I judged that for the OP *finite* only refers to the extension as opposed to *field*, so I removed the *finite-fields* tag. – 2012-11-07