Splitting FieldEdit

A splitting field is a central construct in algebra that captures where a polynomial can be completely factored into linear factors. Given a polynomial f in a base field F, a splitting field is an extension E of F in which f splits as a product of linear polynomials, and E is generated by the roots of f. In more down-to-earth terms, it is the smallest field you need to adjoin all the roots of f so that the polynomial can be written as a product of terms like (x − α) with α running over the roots.

Splitting fields sit at the heart of Galois theory, where the symmetries among the roots—the automorphisms of the splitting field that fix the base field—reveal why some polynomials can be solved by radicals and others cannot. The passage from the algebraic data of a polynomial to the action of a group on its roots is one of the cleanest illustrations of how structure in algebra emerges from simple questions about equations.

Definition and existence

Let F be a field and f ∈ F[x] a non-constant polynomial. A splitting field E of f over F is a field extension E/F such that:

f factors completely into linear factors over E, i.e., f(x) = c ∏(x − αi) with αi ∈ E, and
E is generated by the roots, meaning E = F(α1, α2, ..., αr) where α1, ..., αr are the roots of f in some algebraic closure of F.

Because the roots lie in an algebraic closure, adjoining finitely many roots to F yields a finite extension, so the splitting field exists and is unique up to F-isomorphism.

If the polynomial is separable (no repeated roots), the splitting field is particularly well-behaved: the extension is normal and separable, and the associated Galois group acts faithfully on the set of roots.

Basic properties

The splitting field is the smallest field over which f splits completely; any other field extension in which f splits contains it.
The degree [E : F] is finite for a polynomial of finite degree, since E is obtained by adjoining finitely many roots.
If f is irreducible over F and separable, then any root α generates a finite extension F(α) whose degree equals the degree of f, and the full splitting field is obtained by adjoining all conjugates of α.

These ideas tie into broader concepts in field extension theory and in the study of how polynomials behave under base-field changes.

Examples

Over the rational numbers Q, the polynomial x^2 − 2 has roots ±√2. Its splitting field is Q(√2). The Galois group Gal(Q(√2)/Q) has order 2 and acts by swapping the two roots.
Over Q, the polynomial x^3 − 2 has roots ∛2, ∛2 ω, ∛2 ω^2, where ω is a primitive cube root of unity. Its splitting field is Q(∛2, ω). The Galois group is of order 6 and is isomorphic to the symmetric group S3, reflecting the full permutation symmetry among the three roots.
Over a finite field, say GF(p), the splitting field of a polynomial f ∈ GF(p)[x] is a finite extension GF(p^n) in which all roots lie; in finite fields the structure is particularly tidy because every finite extension is again a finite field.

Galois group and the fundamental correspondence

The Galois group of a splitting field E over F, denoted Gal(E/F), consists of all field automorphisms of E that fix F pointwise. This group encodes how the roots can be permuted while respecting the base field's structure. In the separable case, Gal(E/F) acts by permuting the roots of f, yielding a faithful embedding into the symmetric group on the roots.

A central pillar of the subject is the fundamental theorem of Galois theory, which sets up a correspondence between subgroups of Gal(E/F) and intermediate fields between F and E. Normal subgroups correspond to normal (or Galois) subextensions, and the degree of the fixed-field extension matches the index of the subgroup. This correspondence provides a bridge from the algebraic world of fields to the combinatorial world of groups, and it is precisely this bridge that illuminates why certain polynomials can be solved by radicals and others cannot.

Separable vs inseparable and solvability

Polynomials in characteristic zero are always separable, which makes the splitting field a Galois extension. In positive characteristic, inseparability can occur, complicating the picture, but the basic framework still illuminates the relationship between roots, symmetry, and field structure. The question of solvability by radicals—whether the roots can be expressed using roots of the base field and field operations—reduces, via Galois theory, to whether the Galois group is a solvable group. If the group is solvable, there is a sequence of labeled radical extensions building up the splitting field; if not, radicals alone cannot capture all roots.

Computation and applications

Computing splitting fields typically involves constructing the field by adjoining one or more roots and then iterating to include all conjugates. Techniques include the use of resultants to detect common roots, explicit adjunction of roots, and, in the modern era, computer algebra systems that implement algorithms for factoring polynomials, building normal closures, and determining Galois groups. Splitting fields appear throughout number theory, algebraic geometry, and algebraic coding theory. In the finite-field setting, understanding splitting fields of polynomials over GF(q) informs the design of error-correcting codes and cryptographic constructions, where the interaction between field extensions and polynomial roots underpins practical protocols.