Finite vector-space dimension forces every field element to satisfy a polynomial.Suggest a slogan
Let \(L/K\) be a field extension with \([L:K]<\infty\). Then every \(\alpha\in L\) is algebraic over \(K\).
Let \(n=[L:K]\). The \(n+1\) elements \(1,\alpha,\ldots,\alpha^n\) of the \(n\)-dimensional \(K\)-vector space \(L\) are linearly dependent, so there are \(c_0,\ldots,c_n\in K\), not all zero, with \(\sum_{i=0}^n c_i\alpha^i=0\). Thus \(\alpha\) is a root of the nonzero polynomial \(\sum_{i=0}^n c_ix^i\in K[x]\).