The degree $[K:F]={\dim }_{F}K$ of a finite field extension

Depends on

• Field extensions, generated subrings $F[S]$, generated subfields $F(S)$, and simple extensions
• Finite-dimensional vector space, and its dimension $\dim_F V$; infinite-dimensional means having no finite basis

Used by

• [ℚ(ζₙ):ℚ]=φ(n) and Gal(ℚ(μₙ)/ℚ)≅(ℤ/n)^× Corollary
• A finite purely inseparable extension in characteristic p has degree a power of p Corollary
• Algebraic Bezout formula as a sum of local scheme lengths Corollary
• Every finite extension of a finite field is simple Corollary
• For a finite extension, [K:F]ₛ≤ [K:F] Corollary
• For an odd prime p, ℚ(ζₚ) has exactly one intermediate field of degree two over ℚ Corollary
• Index of a central division algebra Definition
• Normal bases of a finite Galois extension Definition
• Number field Definition
• p-bases for finite exponent-one purely inseparable extensions Definition
• The inseparable degree [K:F]ᵢ=[K:F]/[K:F]ₛ of a finite extension Definition
• The norm N_K/F and trace Tr_K/F of a finite field extension Definition
• The separable degree [K:F]ₛ as a count of embeddings into an algebraic closure Definition
• Total length of a zero-dimensional projective scheme Definition
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A normal basis of F₈ over F₂ Example
• Finite field extensions and etaleness Example
• Gal(F₈/F₂) is cyclic of order three with no proper intermediate field Example
• The four roots of t⁴+t+1 over F₂ are the Frobenius powers of any one of them Example
• The intermediate fields of F_2¹²/F₂ match the divisors of twelve Example
• FALSE: every basis of a finite field over a subfield is a normal basis False statement
• A finite extension with only finitely many intermediate fields is simple Lemma
• A finite normal extension is separable over its purely inseparable fixed field Lemma
• A finite-type field has finite relative algebraic constants Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Finite purely inseparable rational extensions admit a finite Frobenius envelope Lemma
• Finite-type field extensions with zero Ω Lemma
• For a degree-n extension of a field of order q, the q-power map has order exactly n Lemma
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• For E/F finite Galois and L/F finite inside a common field, [EL:F]=[E:F][L:F]/[E∩ L:F] Lemma
• Integral closure in a purely inseparable rational envelope is finite Lemma
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• Restriction partitions embeddings in a finite tower into extension fibres Lemma
• The eventual Hilbert function of a zero-dimensional projective quotient equals its total length Lemma
• A finite extension has degree one if and only if the two fields are equal Proposition
• For finite subextensions in a common field, [EE':F]≤ [E:F][E':F] Proposition
• Φₙ is irreducible over K exactly when [K(ζₙ):K]=φ(n), exactly when the embedding into (ℤ/n)^× is onto Proposition
• A finite extension of a finite field of order q is Galois with cyclic Galois group generated by x↦ x^q Theorem
• A finite-type domain over a field has finite normalization Theorem
• A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem

…and 15 more results.

Dependency tree · two levels

17 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)