Give a one-sentence synopsis of the proof of Theorem 31.3. 

Data from Theorem 31.3

A finite extension field E of a field F is an algebraic extension of F. 

Proof We must show that for a ∈ E, α is algebraic over F. By Theorem 30 .19 if [E : F] = n, then 1, α,··· ,αⁿ cannot be linearly independent elements, so there exist a_i ∈ F such that a_nαⁿ + · · · + a₁α + a₀ = 0, and not all a_i = 0. Then f(x) = a_nxⁿ + ····+ a₁x + a₀ is a nonzero polynomial in F[x]. and f (a) = 0. Therefore, α is algebraic over F.