- #1

Firepanda

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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, x

^{2}, ..., x

^{2n}

Not sure what to do with all this information though.

Any help would be appreciated. Thanks