Tower Law in Field Extensions | Degree Multiplication Theorem

For a Galois extension K/F, the order of the Galois group equals the degree of the extension [K:F]. This is proven by showing that the Frobenius automorphism σ(x) = x^p has order equal to the degree of the extension. For finite fields, this means the Galois group of Z_{p^n}/Z_p has order n, and is isomorphic to Z_n.

Galois theory establishes a fundamental connection between field extensions and group theory, where the degree of a field extension corresponds to the order of its Galois group, and the structure of the Galois group determines whether polynomial equations can be solved by radicals. For example, the field extension generated by cos(5π/12) has degree 4, and its Galois group is isomorphic to the cyclic group of order 4, which explains why this trigonometric value can be expressed using nested square roots.

This segment covers advanced field theory concepts: (1) The field extension ℚ(√2 + √3) over ℚ has degree 4, proven by showing √2 + √3 is not in ℚ, not in ℚ(√2), and not in ℚ(√3). The degree is 4 because [ℚ(√2 + √3) : ℚ] = [ℚ(√2 + √3) : ℚ(√2)] × [ℚ(√2) : ℚ] = 2 × 2 = 4. (2) The field ℚ(√2 + √3) equals ℚ(√2, √3), proven by showing √2 + √3 and √3 - √2 are both in ℚ(√2 + √3), and their sum is 2√3 and difference is 2√2. (3) The Galois group of x⁴ - 10x² + 1 over ℚ is isomorphic to the Klein four-group V₄ (Z₂ × Z₂), with four automorphisms: identity, σ (maps √2 to -√2), τ (maps √3 to -√3), and στ (maps both to negatives). The roots are √2 + √3, -√2 + √3, √2 - √3, and -√2 - √3.

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 degree of a field extension [K:F] equals the product of the degrees of intermediate extensions, and the Galois group of a splitting field is determined by the structure of the polynomial's roots and their symmetries; for example, the splitting field of x^n - 1 over Q has degree φ(n) and a cyclic Galois group isomorphic to (Z/nZ)^×.

A finite field extension E of F is normal if and only if: (1) E is a splitting field of some polynomial over F; (2) Every F-automorphism of any extension L of E sends E to itself; (3) For every β in E, the minimal polynomial of β over F decomposes completely into linear factors in E[x]. The restriction map from F-linear automorphisms of L to F-linear automorphisms of E is a group homomorphism whose kernel consists of E-linear automorphisms of L. When L is also normal over F, this restriction map is surjective. For a tower F ⊆ E ⊆ L, if L is normal over F, then L is normal over E. However, normality does not behave transitively: normality of the top extension does not imply normality of intermediate extensions, and normality of intermediate extensions does not imply normality of the top extension. For any finite field extension E of F, there exists a smallest normal extension of F containing E, called the normal closure, which is unique up to isomorphism and can be constructed as the splitting field of the product of the minimal polynomials of a basis of E over F.

Finite extensions compose transitively: if K ⊂ F and F ⊂ E are both finite, then K ⊂ E is finite. Similarly, algebraic extensions compose transitively: if K ⊂ F and F ⊂ E are both algebraic, then K ⊂ E is algebraic. An element a ∈ F is algebraic over K if and only if there exists an intermediate field E containing a such that K ⊂ E is finite. These properties form the foundation for understanding tower laws and composition of field extensions.

A field extension L/K is normal if and only if any of four equivalent conditions hold: (1) L is the splitting field of some polynomial over K; (2) For every irreducible polynomial f(x) ∈ K[x] with a root in L, all roots lie in L; (3) |Gal(L/K)| = [L:K]; (4) The fixed field of Gal(L/K) is K. Normal extensions are transitive: if L/F and F/K are normal, then L/K is normal. However, normality is not transitive in the sense that if K/F and F/E are normal, K/E may not be normal.

Algebraic extensions compose transitively: if L/K and M/L are algebraic, then M/K is algebraic. If K(α₁), ..., K(αₙ) are finite algebraic extensions of K, then their composite field K(α₁, ..., αₙ) is also a finite algebraic extension of K. This follows from the tower law and the fact that finite extensions are algebraic. Every finite field extension is algebraic, meaning if [L:K] is finite, then every element in L is algebraic over K.

In field theory, a polynomial is separable if its derivative is non-zero, meaning it has no repeated roots in its splitting field; an algebraic extension is separable if every element is separable over the base field, and a field is perfect if all irreducible polynomials are separable (which holds for all fields of characteristic zero). An extension is normal if any irreducible polynomial over the base field that has a root in the extension splits completely into linear factors. For finite extensions, these concepts are characterized by equivalent conditions involving homomorphism extensions and splitting fields.

A finite field (Galois field) with characteristic p (a prime number) must have exactly p^n elements for some integer n ≥ 1, where p is the characteristic and n is the degree of the field extension over its prime subfield. This is proven by showing that the field is a finite-dimensional vector space over its prime subfield (isomorphic to Z_p), and each element can be uniquely expressed as a linear combination of basis elements with coefficients from the prime subfield, resulting in p choices for each of the n basis positions.

This comprehensive section presents the complete classification of finite fields. It begins by establishing that any finite field has order p^n for some prime p and natural number n. The video then develops the theoretical foundation through separable polynomials, proving that a polynomial is separable if and only if it is relatively prime to its derivative. This leads to the existence proof: the polynomial x^(p^n) - x is separable (derivative equals 1) and has exactly p^n distinct roots in its splitting field, forming a subfield closed under field operations. The uniqueness follows from showing any field of order p^n must contain all these roots. The section concludes by analyzing subfield structure, proving that every subfield of GF(p^n) is isomorphic to GF(p^m) for some m dividing n, and conversely, if m divides n, GF(p^n) contains a unique subfield isomorphic to GF(p^m). This creates a lattice structure where subfields correspond to divisors of n.

The Galois theory of finite fields is completely understood: every finite field is uniquely determined by its characteristic (a prime p) and its cardinality (a power of p), and the Galois group of a finite field extension F_p^n/F_p is cyclic of order n, generated by the Frobenius automorphism that maps each element to its p-th power; this cyclic structure extends to an inverse system where the Galois groups of finite subfields form a profinite group isomorphic to the profinite completion of the integers, Ẑ.

For finite fields, F_p^d is a subfield of F_p^n if and only if d divides n. This follows from the polynomial divisibility condition: x^d - 1 divides x^n - 1 if and only if d divides n. This provides a complete characterization of the lattice of subfields of finite fields. The proof uses polynomial division: writing n = qd + r, x^n - 1 = (x^d - 1)(...) + x^r - 1, so divisibility requires r = 0. This criterion also provides a clever proof of Wilson's theorem: for an odd prime p, (p-1)! ≡ -1 mod p. The product of all non-zero elements in F_p is -1, which matches the constant term of the polynomial whose roots are the non-zero elements.

Finite fields have characteristic p (prime) and order p^k. They are unique up to isomorphism and exist as splitting fields of x^(p^k) - x over Z_p. The multiplicative group is cyclic of order p^k - 1, generated by a primitive element. Construction uses Z_p[x]/(f(x)) for irreducible f of degree k. The Galois group of F_{p^k}/F_p is cyclic of order k, generated by the Frobenius automorphism φ(a) = a^p. Conjugates of α are α^(p^i) for i = 0, ..., k-1. Intermediate fields correspond to divisors of k via the Galois correspondence. The structure of intermediate fields mirrors the divisor lattice of k.

The Primitive Element Theorem states that any finite separable extension is a simple extension. For any finite separable extension K/F, there exists γ such that K = F(γ). The proof involves showing that for any two elements α, β, there exists γ = α + aβ such that K = F(γ).

The primitive element theorem states that finite separable extensions have primitive elements. For prime degree extensions, the tower law eliminates intermediate fields, forcing simplicity. If [K:F] = p (prime), then any intermediate field must be trivial, so K = F(α) for some α. This provides a complete characterization of simple extensions among finite extensions.

The primitive element theorem states that finite separable extensions can be generated by a single element α, so E = F(α). For multiple radical extensions, multiply individual degrees: [ℚ(√2, √3, √5):ℚ] = 2 × 2 × 2 = 8. For nested radicals like ℚ(∛2, √2), the degree is 3 × 2 = 6. These calculations determine the vector space dimension of the extension field over the base field.

This section develops the primitive element theorem and its consequences. The theorem states that any finite separable extension F/K is simple, meaning F = K(α) for some element α. The minimal polynomial of α over K is the unique monic irreducible polynomial in K[x] that has α as a root. The section proves that K(α) is isomorphic to K[x]/(m(x)), where m(x) is the minimal polynomial. This construction provides an abstract way to build field extensions from irreducible polynomials, independent of any specific embedding into a larger field. The section also covers field homomorphisms, noting that they are necessarily injective because the kernel is an ideal, and the only ideals in a field are {0} and the field itself.

The Primitive Element Theorem states that for a finite separable extension K/F, there exists a single element α such that K = F(α). This theorem holds when the characteristic of F is 0 or when F is a finite field. For infinite fields of positive characteristic, additional conditions are needed. The theorem simplifies the study of finite extensions by reducing them to simple extensions generated by a single element.