Separable Field Extensions | Algebraic Field Theory Explained

The Fundamental Theorem of Galois Theory states that for a finite Galois extension K ⊂ L (where the Galois group has order equal to the degree of the extension), there is a one-to-one correspondence between subfields M with K ⊂ M ⊂ L and subgroups of the Galois group. This correspondence allows one to determine all intermediate fields by studying the subgroup structure of the Galois group.

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.

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).

The Fundamental Theorem of Galois Theory establishes a bijective correspondence between intermediate field extensions of a Galois extension K/F and subgroups of the Galois group Gal(K/F). For any intermediate field L, the corresponding subgroup consists of all automorphisms in Gal(K/F) that fix every element of L. Conversely, for any subgroup H of Gal(K/F), the fixed field of H (elements fixed by all automorphisms in H) is an intermediate field. The degree [K:L] equals the index [G:H] of the subgroup in the Galois group. Normal subgroups correspond to normal extensions.

To prove the Primitive Element Theorem, it suffices to show that for any two elements α, β in a finite extension K/F, there exists γ such that K = F(γ). The proof constructs γ = α + cβ for some c ∈ F. To show F(α, β) = F(α + cβ), it suffices to show β ∈ F(α + cβ). Consider the minimal polynomial m_β(x) of β over F and the polynomial m_α(x) - c(x - α) of α + cβ. If β is the only common root of these polynomials, then β lies in F(α + cβ). For the polynomials to have only β as common root, we need c ≠ (α_i - α_j)/(β_i - β_j) for all conjugates α_i of α and β_i of β. Since there are finitely many forbidden values and F is infinite (characteristic zero), we can always choose c avoiding all forbidden values. Irreducible polynomials over characteristic zero fields have no multiple roots, which is essential for the proof.

The Primitive Element Theorem states that any finite separable extension has a primitive element α such that E = F(α). For ℚ(√2, √3)/ℚ, √2 + √3 is a primitive element with minimal polynomial x⁴ - 10x² + 1 (degree 4). For ℚ(∛2, ∛3)/ℚ, ∛2 + ∛3 is a primitive element with minimal polynomial x⁹ - 15x⁶ - 36x³ - 1 (degree 9). For ℚ(∛2, ∛3, ∛5)/ℚ, the degree is 27 (3 × 3 × 3). The general method involves identifying minimal polynomials, checking algebraic independence, and applying the Tower Law to compute extension degrees systematically.

The primitive element theorem states that a finite separable extension has a primitive element. For Q(√5, √3), we show Q(√5, √3) = Q(√5 + √3). Given α = √5 + √3, we express √5 = (α² + 8)/(2α) and √3 = α - √5. Since these expressions use only field operations on α and rational numbers, both √5 and √3 belong to Q(α). Conversely, α = √5 + √3 clearly belongs to Q(√5, √3). Thus the fields are equal.

The Primitive Element Theorem states that any finite extension of the rational numbers can be generated by a single element, meaning that even complex extensions involving multiple non-rational roots (such as Q(∛2, ∛3)) can be expressed as a simple extension generated by a single primitive element (like γ = ∛2 + ∛3), which can be verified through linear algebra techniques and computational tools like Sage.

A primitive field extension is defined abstractly as the factor field by the ideal generated by an irreducible polynomial P. If P has a root α in some field C, then the extension can be constructed by adjoining α to the base field K. Elements in this extension are polynomials in α of degree strictly less than the degree of P. The Primitive Element Theorem states three key properties: (1) The number of embeddings of L into C over K is at most [L:K]; (2) If we consider embeddings into C specifically, there are exactly [L:K] such embeddings; (3) There exists an element α in C such that L = K(α), meaning the extension is primitive. To prove this for composite extensions K(α, β), we use the tower of fields technique. Since [L:K] = N, there are N embeddings of L into C over K. Each embedding is uniquely determined by its values on α and β. Since K is infinite (containing Q), we can find an element x in K such that the ratio f(α)/f(β) differs from g(α)/g(β) for any other embedding g. This ensures that the constructed element γ = α + xβ generates the entire field L. Two fields are equal if and only if they have the same degree over the base field. An automorphism of a field L over K is a bijective map from L to L that fixes K pointwise and preserves field operations. The number of such automorphisms is at most [L:K], with equality if and only if the extension is a Galois extension. A splitting field (or field of decomposition) of a polynomial P over K is the smallest field extension of K in which P factors completely into linear factors. This is equivalent to adjoining all roots of P to K. The degree of a splitting field L over K for a polynomial of degree n is at most n!. For the polynomial x³ - 2 over Q, the splitting field is Q(∛2, ω), where ω = e^(2πi/3) is a primitive cube root of unity. The degree of this extension is 6, not 3, because while ∛2 generates a cubic extension, we must also adjoin ω to split the polynomial completely. The discriminant of a polynomial P(x) = aₙxⁿ + ... + a₀ is defined as the product of squared differences of its roots: Δ = ∏(αᵢ - αⱼ)². For a cubic polynomial x³ + px + q, the discriminant is -4p³ - 27q². For an irreducible cubic polynomial over Q, the Galois group is either S₃ (order 6) or A₃ (order 3). The group is S₃ if the discriminant is not a square in Q, and A₃ if the discriminant is a square. A finite field extension L/K is Galois if and only if any of the following equivalent conditions hold: (1) L is the splitting field of some polynomial over K; (2) L is a primitive extension K(α) where the minimal polynomial of α splits completely into linear factors over L; (3) The number of automorphisms of L over K equals [L:K]. A field extension L/K is normal if every irreducible polynomial over K that has a root in L splits completely in L. The Primitive Element Theorem allows us to prove that any element in a Galois extension satisfies the 'star property' (its minimal polynomial splits completely in L). For a primitive extension L = K(α), the automorphisms of L over K are in one-to-one correspondence with the roots of the minimal polynomial of α.

Inseparable extensions fundamentally differ from separable ones—they may lack primitive elements entirely. The example K = F_p(t^p, u^p) ⊂ M = F_p(t, u) demonstrates this: every element satisfies x^p ∈ K, making the extension inseparable of degree p^2. Critically, this extension contains infinitely many intermediate subfields, violating the finite intermediate extension property that characterizes separable extensions.

Finite field extensions are always separable because their elements satisfy polynomials like x^{p^n} - x with distinct roots. Purely inseparable extensions occur when every element satisfies x^{p^n} = a for some a in K, yielding only one distinct root. Any algebraic extension decomposes into a tower: first a separable extension, then a purely inseparable extension. Classical Galois theory focuses on separable extensions because non-separable extensions have trivial Galois groups. While generalizations exist using algebraic group schemes, standard Galois theory requires discrete Galois groups, making separable extensions central to the theory.

For a separable extension L/K, the set LS of all separable elements of L over K generates a separable extension LS/K. If L ≠ LS, then L/LS is purely inseparable. This means any element of L not in LS is purely inseparable over LS. The purely inseparable degree [L:LS] is called the inseparable degree of L/K. This result characterizes the structure of separable extensions by identifying the maximal separable subextension and the purely inseparable part.

This section proves that for a primitive extension F = K(α) with minimal polynomial m(x), the number of embeddings of F into C that fix K equals the degree of m(x), which is [F:K]. This follows because the number of roots of m(x) in C equals its degree (for separable polynomials), and each root gives a distinct embedding. The section distinguishes between separable and inseparable extensions: in characteristic 0, all irreducible polynomials are separable (no multiple roots), while in characteristic p > 0, irreducible polynomials may have multiple roots (inseparable). For example, x^p - a over F_p(a) is irreducible but has only one root with multiplicity p. The primitive element theorem fails for inseparable extensions, and the number of embeddings is strictly less than the degree.

Inseparable extensions do occur in fields of positive characteristic. For example, consider the field extension E = Z_2(t) (rational functions in t over Z_2) over F = Z_2(t^2). The polynomial x^2 - t^2 is irreducible over F but factors as (x-t)^2 in E, showing it has a repeated root t. This demonstrates that in characteristic 2, extensions can fail to be separable. Similar examples exist for any prime characteristic.