Field extension that is not normal

[jostpuur]

I want to come up with an example of a field extension that is not normal, and seems to be difficult. All extension constructed in some obvious way tend to turn out normal.

 

[micromass]

I don't know how you defined normal exactly (there are various equivalent definitions). But a very useful characterization is the following: if $K$ is normal over $L$ and if $P(X)$ is a irreducible polynomial over $L$ that has a root in $K$, then $P(X)$ has all roots in $K$.

So take $L=\mathbb{Q}$. Take some irreducible polynomial over $\mathbb{Q}$. Adjoin a root of this polynomial to $\mathbb{Q}$ and see whether all roots are included.

For examply, we know that $X^3 - 2$ is irreducible over $\mathbb{Q}$. Do all roots of $X^3-2$ lie in $\mathbb{Q}[\sqrt[3]{2}]$?

 

[jostpuur]

oh dear...