I think these are related:
Definition. Let F be an extension field of K and let u be in F. If there exists a nonzero polynomial f(x) in K[x] such that f(u)=0, then u is said to be algebraic over K. If there does not exist such a polynomial, then u is said to be transcendental over K.
Proposition. Let F be an extension field of K and let u in F be an element algebraic over K. If the minimal polynomial of u over K has degree n, then K(u) is an n-dimensional vector space over K.
Definition. Let F be an extension field of K. If the dimension of F as a vector space over K is finite, then F is said to be a finite extension of K.
The dimension of F as a vector space over K is called the degree of F over K, and is denoted by [F:K].
Proposition. Let F be an extension field of K and let u be in F. The following conditions are equivalent:
(1) u is algebraic over K;
(2) K(u) is a finite extension of K;
(3) u belongs to a finite extension of K.
What I know:
I've been staring at [K(x):K] = 2n+1 , n>=0, for a while now.
So this means the degree of the minimal polynomial is odd
A basis of K(x) over K is 1, x, x2, ..., x2n
Not sure what to do with all this information though.
Any help would be appreciated. Thanks