Alphabeta Math

Algebraic Closure, Embeddings, and Separability — Examples

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 11 results · all verified · 11 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 11 also cleared it.
1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Assuming Choice, real algebraic numbers embed properly in an algebraic closure of ℚ

Example

Fₚ is the union of its finite subfields and is an infinite algebraic extension

Example

ℚ(³√2) has three embeddings into ℚ̄ but only one ℚ-automorphism

Example

ℚ(√2,√3) has four embeddings into ℚ̄

Example

xᵖ-t is irreducible and inseparable over Fₚ(t)

Counterexample

Fₚ(s,t)/Fₚ(sᵖ,tᵖ) has degree p², infinitely many intermediate fields, and no primitive element

Example

⋃_n≥0Fₚ(t^1/pⁿ) is an infinite perfect field of characteristic p

False statement

FALSE: every irreducible polynomial over a field is separable

False statement

FALSE: every finite extension satisfies [K:F]ₛ=[K:F]

False statement

FALSE: every algebraic extension is simple

False statement

FALSE: an algebraic closure is unique up to a unique base-field isomorphism

Sources