How Do Degree Odd Polynomials Relate to Extension Fields of K?

[kathrynag]

1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2.

2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2).



1. I was thinking of doing something like [K(u):K(a)][K(a):K(a^2)]

2. algebraic so there exists a polynomial m(x) such that m(a)=0.
Just not sure how to work in the degree being odd.

 

[kathrynag]

Is this a good way to start or should I try something different?

 

[kathrynag]

2. I was thinking of somehow using a theorem stating [F:K]=[F:K(u)][K(u):K]
based on the deg m(x) being odd, I would say deg m(x)=2n+1