Splitting Fields in Field Extensions | Abstract Algebra

A field extension $ K \subseteq L $ is called separable if every element of $ L $ is a root of a separable polynomial over $ K $, meaning polynomials without multiple roots; in characteristic zero, all extensions are automatically separable, but in positive characteristic $ p $, inseparable extensions can occur where elements satisfy equations like $ x^p = a $, and any algebraic extension can be decomposed into a separable extension followed by a purely inseparable extension.

A separable extension is a field extension where the degree equals the separable degree. The term 'separable' means 'separated' or 'distinct' - roots should be distinct and not repeated. This concept is fundamental in Galois theory and relates to whether a polynomial has multiple roots.

If k is the fixed field of the Galois group, then M is a splitting field of a separable polynomial over k. For any element α in M, the minimal polynomial of α has distinct roots (separable) and is fixed by the Galois group, so its coefficients lie in k. Taking the product of such polynomials for all generators gives a separable polynomial whose splitting field is M.

A splitting field of a polynomial f(x) over K is the smallest field extension containing all roots of f(x). Examples include Q(√2) for x²-2 and Q(√3, √5) for x⁴-8x²+15. A normal extension F/K is one where every irreducible polynomial in K[x] with one root in F splits completely in F. A key theorem states that F/K is a splitting field if and only if it is a finite normal extension. A polynomial is separable if it has n distinct roots (no multiple roots). An element α is separable if its minimal polynomial is separable. An extension is separable if all its elements are separable. A crucial theorem states that all algebraic extensions of fields with characteristic 0 are separable. Since Q has characteristic 0, all algebraic extensions of Q are automatically separable.

An extension is separable if all elements have minimal polynomials with no repeated roots. For finite fields, all extensions are separable because the characteristic is p and the Frobenius map is injective. The splitting field of a polynomial contains all its roots, and if an extension is a splitting field, it is automatically normal. These properties together determine whether an extension is Galois.

The Fundamental Theorem of Galois Theory establishes a bijective correspondence between subfields of a finite separable normal field extension K/F and subgroups of its Galois group G = Gal(K/F), where the correspondence is inclusion-reversing: larger subgroups correspond to smaller subfields. The Galois group consists of all field automorphisms of K that fix every element of F, and its order equals the degree of the extension when the extension is Galois (normal and separable). This theorem provides a powerful connection between group theory and field theory, allowing us to study field extensions through their symmetry groups.

Galois theory revolutionized mathematics by establishing deep connections between field extensions and group theory. The Fundamental Theorem of Galois Theory creates a correspondence between intermediate fields and subgroups of the Galois group. A polynomial is solvable by radicals if and only if its Galois group is solvable—a group that can be broken down into abelian factors. Since symmetric groups Sₙ for n ≥ 5 are not solvable, quintic and higher-degree polynomials generally lack solution formulas. This elegant framework translates complex field-theoretic questions into more manageable group-theoretic ones, representing one of mathematics' greatest achievements.

This section presents the central result of Galois theory. It introduces automorphisms (field rearrangements preserving addition and multiplication) and defines the Galois group as automorphisms that leave a subfield unchanged. The key insight is that Galois groups permute the roots of polynomials: if x is a root, then σ(x) is also a root for any automorphism σ. The section establishes the subgroup relation for nested field extensions and proves that the quotient of Galois groups is isomorphic to the Galois group of the intermediate field. The fundamental theorem is then stated: there is a one-to-one correspondence between fields in a field extension and subgroups of its Galois group, with the lattices being the same except one is upside down. This correspondence allows translating field extension questions into group theory questions.

The Fundamental Theorem of Galois Theory establishes a one-to-one correspondence between the intermediate fields of a finite normal extension K/F and the subgroups of the Galois group G(K/F), where each intermediate field E corresponds to the subgroup G(K/E) and vice versa, with additional properties including: the degree of the extension [K:E] equals the order of G(K/E), the fixed field of a subgroup H is K^H, G(K/E) is a normal subgroup of G(K/F) if and only if E is a normal extension of F, and the quotient group G(K/F)/G(K/E) is isomorphic to G(E/F).

Galois Theory establishes a fundamental correspondence between field extensions and group theory, demonstrating that the structure of a Galois extension (algebraic, separable, and normal) is completely determined by its Galois group. This powerful framework, developed by Évariste Galois, provides a systematic method for determining whether polynomial equations can be solved by radicals, proving that general polynomial equations of degree 5 or higher cannot be solved using radicals. The theory reveals that the solvability of polynomial equations depends on whether their Galois group is a solvable group, with applications including the constructibility of regular polygons and the properties of cyclotomic fields.

A finite field F with q elements is the splitting field of the polynomial x^q - x over its prime subfield, because all q elements of F are roots of this polynomial: for any element a in F, a^q = a (with a^q - a = 0), and since the polynomial has degree q and F contains exactly q elements, all roots must belong to F.

Finite fields are classified by their order, which must be a prime power p^n for some prime p and integer n ≥ 1, and for each prime power there exists a unique finite field up to isomorphism; this field can be constructed as the splitting field of the polynomial x^p^n - x over the prime field F_p, or explicitly by adjoining a root of an irreducible polynomial of degree n over F_p, though no canonical choice of irreducible polynomial exists for a given degree and field characteristic.

Every finite field of order q=p^n is a splitting field for the polynomial x^q-x over the prime field F_p, which proves that all finite fields of the same order are isomorphic to each other.

Every finite field with q elements is the splitting field of the polynomial f(x) = x^q - x over its prime subfield, meaning all elements of the field are roots of this polynomial, and the field is the smallest extension containing all these roots.

For every prime power q = p^n, there exists a finite field with exactly q elements. The proof constructs this field as a splitting field of x^q - x over the prime field F_p, or equivalently as F_p[x]/(f(x)) for an irreducible polynomial f of degree n. Up to isomorphism, there is exactly one finite field for each prime power. The Galois group of a finite field extension is cyclic and generated by the Frobenius automorphism. This classification theorem is fundamental for understanding the structure of finite fields and their extensions.

An algebraic closure of F is an extension F̄ that is algebraic over F and contains all roots of all polynomials over F. A field K is algebraically closed if every polynomial with coefficients in K has a root in K (which implies it splits completely). An algebraic closure of an algebraically closed field is the field itself. For any field F, there exists an algebraically closed field K containing F. The algebraic closure of F is the subfield of K consisting of all elements algebraic over F, and is unique up to isomorphism. The Fundamental Theorem of Algebra states that C is algebraically closed, so Q̄ (complex numbers algebraic over Q) is an algebraic closure of Q.

An algebraic closure of a field F is an algebraic extension K of F such that every polynomial in F[x] splits completely into linear factors in K[x]. This means K contains all roots of all polynomials with coefficients in F. The algebraic closure is unique up to isomorphism.

A field F is algebraically closed if every non-constant polynomial in F[x] has a root in F, equivalently factors completely into linear factors. Examples: Q is not algebraically closed (x²-2 has no root), C is algebraically closed (fundamental theorem of algebra). An algebraic closure of F is an extension E ⊇ F such that E is algebraic over F and E is algebraically closed. The algebraic closure of R is C (degree 2), while the algebraic closure of Q is the field of algebraic numbers (countable), distinct from C which is uncountable.

An algebraic closure of a field K is a field L containing K such that L is algebraic over K and every non-constant polynomial with coefficients in K has a root in L. This concept extends the field to include all algebraic roots of polynomials.

A field K is algebraically closed if and only if any of the following equivalent conditions hold: (1) Every non-constant polynomial in K[x] has a root in K, (2) Every non-constant polynomial in K[x] factors completely into linear factors in K[x], (3) K has no non-trivial finite extensions. An algebraic closure of a field K is an algebraic extension L of K that is algebraically closed. For any field K, there exists an algebraic extension K₁ of K with the property that every non-constant polynomial in K[x] has a root in K₁. This is constructed by repeatedly adjoining roots: K₀ = K, K₁ = K₀(α₁) where α₁ is a root of some polynomial in K₀[x], K₂ = K₁(α₂) where α₂ is a root of some polynomial in K₁[x], and so on. The union L of all Kₙ is a field, and every element of L is algebraic over K.