Thread: Splitting Field

1. Prove that if F is an extension field of K of degree 2, then F is the splitting field over K for some polynomial.



ideas:
I found a corollary in my book saying if F is a finite extension of K, and u in F then the degree of u over K is a divisor of [F:K]. I have a feeling that may help.
We have a splitting field if [F:K]<=n! with n being the degree.
So we want to show we have [F:K]<=2 since 2 was the degree.

suppose F is an extension field of K of degree 2. this means that F is a K-vector space of dimension 2. so we can find a u in F with {1,u} as a basis. this means that

u^2 (which is in F), can be expressed as u^2 = au + b, for some a,b in K. so u is a root of x^2 - ax - b in K[x], and -u is another root, meaning K is a splitting field

of x^2 - ax - b.