Is the field extension normal?

[futurebird]

I've been asked to find out if some field extensisons are normal. I want to know if I'm thinking about these in the right way.

For Q(a):Q

I first find the minimal polynomial for a in Q[a]. Then I look at all zeros of that polynomial. If all of the zeros are in Q(a) the extension is normal.


Example:

Q(1+i):Q

1+i = x
-1 = x^2-2x+1

x^2-2x+2 is irreducible over Q and the minimal polynomial of 1+i.

the zereos are: 1+i, 1-i

they are both in Q(1+i) so this is a normal extension.


Correct?

 

[futurebird]

*bump*

 

[Dick]

Yes, it's normal. Do you know why field extensions might not be normal?

 

[futurebird]

Q(2^(1/3)):Q is not normal since the minimal polynomial has two imaginary roots that are not in Q(2^(1/3)). Is that the right idea?

 

[futurebird]

I forgot to say that 2^(1/3) is the real cube root of two.

 

[Dick]

Yes, I think so. In your first example, including one root automatically includes the other. In the second it doesn't include the other roots.