Let \(k\) be a field and \(L/k\) a field extension. Then \(L/k\) is finite and normal if and only if \(L\) is the splitting field over \(k\) of some nonzero polynomial \(f\in k[x]\).
Fix an algebraic closure \(\bar k\) containing \(L\).
Suppose \(L/k\) is finite and normal, and write \(L=k(\beta_1,\ldots,\beta_r)\). Let \(f\) be the product of the minimal polynomials of the \(\beta_i\) over \(k\). By normality each factor splits in \(L\), and \(L\) is generated over \(k\) by roots of \(f\), so \(L\) is the splitting field of \(f\).
Conversely, let \(L=k(\alpha_1,\ldots,\alpha_n)\), where \(\alpha_1,\ldots,\alpha_n\) are the roots of \(f\) in \(L\) and \(f\) splits in \(L\). Then \(L/k\) is finite. A \(k\)-embedding \(\sigma\colon L\to\bar k\) sends each \(\alpha_i\) to a root of \(f\), and \(f\) splits in \(L\), so \(\sigma\) permutes \(\theset{\alpha_1,\ldots,\alpha_n}\) and \(\sigma(L)=L\). By PR-OZYUC, \(L/k\) is normal.