A finite field extension K of F is a splitting field if and only if every irreducible polynomial over F has no roots in K or splits over K

Let K be a finite extension of a field F. Prove that K is a splitting field over F if and only if every irreducible polynomial over F with a root in K splits over K.


We begin with some lemmas.

Lemma 1: Let K be a finite extension of F, and let \alpha be algebraic over F. If K is a splitting field for a set S \subseteq F[x], then K(\alpha) is a splitting field for S considered as polynomials over F(\alpha). Proof: Certainly the roots of elements of S are in K(\alpha) \supseteq K. Now suppose E is a splitting field for S over F(\alpha); in particular, E contains F and the roots of the polynomials in S, so K \subseteq E. Moreover \alpha \in E, so K(\alpha) \subseteq E. Hence K(\alpha) is a splitting field for S over F(\alpha). \square

Suppose K is a splitting field for some (finite) set S \subseteq F[x]. Let q(x) be irreducible over F, and suppose \alpha and \beta are roots of q with \alpha \in K. By Theorem 8 on page 519 of D&F, \sigma : \alpha \mapsto \beta extends to an isomorphism \sigma : F(\alpha) \rightarrow F(\beta). Now K(\alpha) is the splitting field for S \subseteq F(\alpha)[x], and likewise K(\beta) is the splitting field for S over F(\beta). By Theorem 27 on page 541 of D&F, the isomorphism \sigma extends to an isomorphism \psi : K(\alpha) \rightarrow K(\beta). We can visualize this using the following diagram.

A field diagram

Since \alpha \in K, [K(\alpha):K] = 1, and so [K(\beta):K] = 1, and we have \beta \in K. That is, if an irreducible polynomial over F has any roots in K, then it splits completely over K.

Conversely, suppose K is finite over F and that every irreducible polynomial with a root in K has all roots in K. Since K is a finite extension, it is algebraic (see Theorem 17 on page 526 of D&F). Say K = F(\alpha_1,\ldots,\alpha_n), and let m_i be the minimal polynomial of \alpha_i over F. By our hypothesis, each m_i splits over K. Moreover, any field over which all the m_i split must contain the \alpha_i. So in fact K is the splitting field over F of the set of polynomials m_i.

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