Splitting fields are the smallest fields where a polynomial breaks into linear factors. Normality is the extension-level condition that says all conjugate roots have been included.
Splitting fields
Definition. Let \(f\in K[x]\). A splitting field of \(f\) over \(K\) is an extension \(L/K\) such that \(f\) factors into linear factors over \(L\), and \(L\) is generated over \(K\) by the roots of \(f\).
Theorem. Every nonconstant polynomial has a splitting field, unique up to \(K\)-isomorphism.
Proof. Existence follows by adjoining one root of an irreducible factor and inducting on the degree. Uniqueness follows by extending a \(K\)-isomorphism step by step along roots of corresponding irreducible polynomials.
Normal extensions
Definition. An algebraic extension \(L/K\) is normal if every irreducible polynomial in \(K[x]\) that has one root in \(L\) splits completely over \(L\).
Theorem. A finite extension \(L/K\) is normal iff \(L\) is the splitting field over \(K\) of some polynomial in \(K[x]\).
Embedding criterion
Proposition. Let \(L/K\) be finite inside an algebraic closure \(\overline K\). Then \(L/K\) is normal iff every \(K\)-embedding \(L\hookrightarrow\overline K\) has image \(L\).
Proof. A \(K\)-embedding sends a root of an irreducible polynomial to another root of the same polynomial. Normality says all those conjugate roots already lie in \(L\), so the embedding lands in \(L\). Conversely, if every embedding preserves \(L\), then conjugates of elements of \(L\) stay in \(L\), forcing splitting.
Quadratic and cubic examples
Example. Every quadratic extension in characteristic not \(2\) is normal: if \(L=K(\sqrt a)\), then the minimal polynomial \(x^2-a\) has roots \(\pm\sqrt a\), both in \(L\).
Counterexample. \(\mathbb Q(\sqrt[3]{2})/\mathbb Q\) is not normal. The polynomial \(x^3-2\) has one root \(\sqrt[3]{2}\) in the field, but the other roots \(\zeta_3\sqrt[3]{2}\) and \(\zeta_3^2\sqrt[3]{2}\) are not real and hence not in \(\mathbb Q(\sqrt[3]{2})\).
Normal closure
Definition. The normal closure of a finite extension \(E/K\) inside \(\overline K\) is the smallest normal extension of \(K\) containing \(E\). It is obtained by adjoining all conjugates of generators of \(E\).
Calculation. The normal closure of \(\mathbb Q(\sqrt[3]{2})\) is \(\mathbb Q(\sqrt[3]{2},\zeta_3)\), the splitting field of \(x^3-2\). This extension has degree \(6\) over \(\mathbb Q\).
Normality is not separability
Remark. Normality says all conjugates are present. Separability says the conjugates are distinct. Galois extensions require both conditions for finite extensions.