embedding into algebraic closure

When you have an abstract field extension, it helps to plant it inside a big ambient field where everything already splits — the algebraic closure. An embedding is exactly such a planting: a way of placing the extension faithfully inside the closure. Counting the different ways to do this is how Galois theory measures the 'size' of an extension's symmetry.

Formally, fix an algebraic closure K-bar of K. For an algebraic extension L/K, a K-embedding of L into K-bar is an injective field homomorphism L -> K-bar that restricts to the identity on K. Field homomorphisms between fields are automatically injective, and any embedding of K into an algebraically closed field extends to any algebraic extension L — the existence resting ultimately on Zorn's lemma. The images of all such embeddings are the conjugate fields of L inside K-bar.

The number of distinct K-embeddings of a finite extension L into K-bar is the separable degree [L : K]_s, which is at most [L : K] with equality precisely when L/K is separable. When L/K is moreover normal (hence Galois in the separable case), every embedding lands in L itself, so the embeddings are exactly the automorphisms and their count equals |Gal(L/K)| = [L : K]. Thus embeddings unify the notions of separability, normality, and the order of the Galois group into one counting principle.