Finite Galois extensions and $\mathrm{Gal}(K\mathrm{/}F)$

Depends on

• Relative field automorphisms and $\operatorname{Aut}(K/F)$
• A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there
• Separable algebraic elements and separable extensions

Used by

• A finite Galois extension is Galois over every intermediate field Corollary
• A cyclic extension is a finite Galois extension with cyclic Galois group Definition
• Galois conjugates of a representation Definition
• Normal bases of a finite Galois extension Definition
• Semilinear Galois actions, twists, and split central idempotents Definition
• The Galois closure of a finite separable extension Definition
• The Galois group of a separable polynomial Definition
• {1+i, 1-i} is a normal basis of ℂ/ℝ while {1,i} is not Example
• A finite normal extension is separable over its purely inseparable fixed field Lemma
• Canonical resolutions over non-algebraically-closed ground fields 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
• Over an infinite base field, no nonzero polynomial vanishes at the conjugate tuple of every element Lemma
• Every intermediate field of ℚ(μₙ)/ℚ is Galois over ℚ with abelian Galois group Proposition
• A finite extension of a finite field of order q is Galois with cyclic Galois group generated by x↦ x^q Theorem
• Equivalent characterizations of a finite Galois extension Theorem
• Every finite abelian group is the Galois group of some finite Galois extension of ℚ Theorem
• Every finite cyclic extension has a normal basis Theorem
• Every finite Galois extension of an infinite field has a normal basis Theorem
• K(μₙ)/K is Galois and σ↦ a_σ embeds its Galois group into (ℤ/n)^× Theorem
• Polynomial algebras over fields have finite integral closures Theorem
• The fundamental theorem of finite Galois theory Theorem

Dependency tree · two levels

8 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

• J. S. Milne, Fields and Galois Theory, v5.10, Definition 3.9 (standard reference, not scraped)
• K. Conrad, The Galois Correspondence, Definition 4.4 (standard reference, not scraped)