State the definition of the splitting field of a polynomial \(f(x) \in F[x]\) over a field \(F\). Prove the existence and uniqueness (up to \(F\)-isomorphism) of splitting fields.
Solution
A splitting field of \(f\in F[x]\) is an extension \(L/F\) such that
- \(f\) splits into linear factors in \(L[x]\), and
- \(L\) is generated over \(F\) by the roots of \(f\).
Existence follows by induction on \(\deg f\). If \(f\) does not already split, choose an irreducible factor \(p\) of degree \(>1\) and form \[ E=F[x]/(p). \] The class of \(x\) is a root of \(p\), hence of \(f\), so over \(E\) one linear factor can be removed. Induction on the remaining degree produces a field in which all roots lie; adjoining exactly those roots gives a splitting field.
For uniqueness, let \(L_1,L_2\) be splitting fields of \(f\) over \(F\). We extend an embedding one root at a time. Suppose \(E\subseteq L_1\) has already been embedded into \(L_2\) by \(\varphi:E\to L_2\), and let \(\alpha\in L_1\) be another root of \(f\). Let \(m_{\alpha,E}(x)\in E[x]\) be its minimal polynomial over \(E\). Since \(m_{\alpha,E}\mid f\) in \(E[x]\), the transported polynomial \(\varphi(m_{\alpha,E})\) divides \(f\) in \(\varphi(E)[x]\). Because \(f\) splits in \(L_2\), this polynomial has a root \(\beta\in L_2\). Since \(E(\alpha)\cong E[x]/(m_{\alpha,E})\) and \(\beta\) is a root of \(\varphi(m_{\alpha,E})\), the map \(\varphi\) extends to \[ E(\alpha)\longrightarrow L_2, \qquad \alpha\longmapsto\beta. \] Starting from \(E=F\) and repeating over the finitely many roots gives an \(F\)-embedding \(L_1\to L_2\). Its image is a splitting field of \(f\) inside \(L_2\); by minimality of \(L_2\), the image is all of \(L_2\). Thus splitting fields are unique up to \(F\)-isomorphism.