Examples of normal and non-normal extensions of Q?

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

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

 

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?

 

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

 

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