Examples of normal and non-normal extensions of Q?

[ae1709]

Hi, I'm really struggling to find examples (with proofs) of the following:

1) For each n>2 give an example of a non-normal extension of Q of degree n.

2) Give examples of normal extensions of Q of degrees 3,4 and 5.

3) Show that for any positive integer n, there exists a normal extension of Q of degree n.

Any help would be much appreciated!

 

[micromass]

So an extension \mathbb{Q}\subseteq K is normal if for all \alpha\in K the minimal polynomial of \alpha splits in K. Or equivalently if K is the splitting field of a polynomial in \mathbb{Q}.

So, can you adjoin a number \alpha to \mathbb{Q} such that the minimal polynomial doesn't split?? This answers (a).

 

[Deveno]

for 2) remember that the algebraic closure of Q is not a subfield of R, so you need to look for some complex numbers that make this happen. i suggest looking on the unit circle, perhaps?

 

[riskos]

So an extension \mathbb{Q}\subseteq K is normal if for all \alpha\in K the minimal polynomial of \alpha splits in K. Or equivalently if K is the splitting field of a polynomial in \mathbb{Q}.

So, can you adjoin a number \alpha to \mathbb{Q} such that the minimal polynomial doesn't split?? This answers (a).

But adjoining one number will surely not give you an extension of degree n as required in the question?

 

[micromass]

But adjoining one number will surely not give you an extension of degree n as required in the question?

It might, for example, adjoining \sqrt[3]{2} gives you a nonnormal extension of degree 3.