Making room for every root
Given a polynomial f ∈ K[x], its splitting field is the smallest extension L/K in which f factors completely into linear factors — and which is generated by those roots. You build it by repeatedly adjoining roots: find an irreducible factor, adjoin a root via K[x]/(that factor), and repeat over the new field until f splits. The splitting field exists for any f, and its degree over K divides (deg f)!.
Splitting field of f(x) = x^3 - 2 over Q. Roots: a = 2^(1/3), a*w, a*w^2 where w = e^(2*pi*i/3) is a primitive cube root of 1. Step 1: adjoin a. [Q(a):Q] = 3 (x^3-2 is Eisenstein at p=2, so irreducible). But Q(a) is REAL, it cannot contain the complex roots a*w, a*w^2. Step 2: adjoin w (root of x^2 + x + 1). [Q(a,w):Q(a)] = 2. Tower law: [Q(a,w):Q] = 3 * 2 = 6. So the splitting field has degree 6, not 3 — splitting needs the roots of unity, not just one real cube root. (Its Galois group is S_3.)
Uniqueness and normality
Splitting fields are unique up to K-isomorphism — any two splitting fields of the same f are isomorphic by a map fixing K. The proof is the central technical lemma of the whole subject: an isomorphism of base fields extends, root by root, to an isomorphism of splitting fields, because each root's minimal polynomial is carried to an irreducible factor on the other side. This extension-of-embeddings argument is worth mastering; it powers everything in Galois theory.
A finite extension L/K is normal when it captures roots in an all-or-nothing way: if an irreducible polynomial in K[x] has *one* root in L, it has *all* its roots in L. The clean theorem: for finite extensions, normal ⟺ L is the splitting field of some polynomial over K. Normality is the “no half-caught roots” condition; you already glimpsed it in Volume I as part of what makes Galois theory work.
The closed universe
Push splitting to its limit and you get an algebraically closed field: one where *every* nonconstant polynomial already factors into linear pieces. ℂ is the famous example — that is exactly the fundamental theorem of algebra. An algebraic closure K̄ of K is an algebraic extension of K that is itself algebraically closed; it is the splitting field of *all* polynomials over K at once.
Two facts to carry: every field has an algebraic closure, and it is unique up to (non-canonical) isomorphism. The existence proof needs Zorn's lemma — be honest that this is genuinely non-constructive — but once you have K̄, every algebraic extension of K embeds into it. That is why people speak of “fixing an algebraic closure” and working inside it: it is the ambient room where all roots live and all embeddings land.