Tower law for finite extensions: $[L:F]=[L:K][K:F]$

Depends on

• Products of bases form a basis in a tower of finite extensions
• The degree $[K:F]=\dim_F K$ of a finite field extension

Used by

• [K:F]=[K:F]ₛ[K:F]ᵢ, and in positive characteristic the inseparable degree is a power of p Corollary
• A finite purely inseparable extension in characteristic p has degree a power of p Corollary
• An algebraically constructible real algebraic number has degree over ℚ equal to a power of two Corollary
• For a finite extension, [K:F]ₛ≤ [K:F] Corollary
• For a finite extension, |Aut(K/F)| divides [K:F] Corollary
• The algebraic numbers in ℂ form an algebraic closure of ℚ Corollary
• The degree of an intermediate field divides the degree of a finite extension Corollary
• ℚ(√2,√3) has degree four and equals ℚ(√2+√3) Example
• ℚ(√2,√3) has four embeddings into ℚ̄ Example
• The complete Galois correspondence for ℚ(√2,√3)/ℚ Example
• The full S₃ correspondence for the splitting field of x³-2 Example
• FALSE: a polynomial solvable by radicals must have abelian Galois group False statement
• FALSE: degrees add in a tower of finite field extensions False statement
• A finite normal extension is separable over its purely inseparable fixed field Lemma
• A finite-type field has finite relative algebraic constants Lemma
• A quadratic extension in characteristic not 2 is obtained by adjoining a square root Lemma
• A simple finite extension has only finitely many intermediate fields Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
• For a degree-n extension of a field of order q, the q-power map has order exactly n Lemma
• For E/F finite Galois and L/F finite inside a common field, [EL:F]=[E:F][L:F]/[E∩ L:F] Lemma
• Separable generation after finite purely inseparable extensions Lemma
• A finite extension is separable if and only if [K:F]ₛ=[K:F] Theorem
• A minimal generating family in a finite exponent-one purely inseparable extension is a p-basis and gives degree pʳ Theorem
• Algebraicity is transitive in towers of field extensions Theorem
• An algebraic extension generated by separable elements is separable Theorem
• An extension generated by finitely many algebraic elements is finite Theorem
• Equivalent characterizations of a finite Galois extension Theorem
• Finitely generated extensions of a perfect field are separably generated Theorem
• Norm is multiplicative, trace is F-linear, and both are transitive in towers Theorem
• Polynomial algebras over fields have finite integral closures Theorem
• ℚ(μₘ)∩ℚ(μₙ)=ℚ(μ_gcd(m,n)) Theorem
• Separability is transitive in towers of algebraic extensions Theorem
• The complex numbers are algebraically closed Theorem
• The fundamental theorem of finite Galois theory Theorem
• The separable degree divides the degree of every finite extension Theorem
• The subfields of F_pⁿ are the unique fields F_pᵈ for positive divisors d of n Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)