Alphabeta Math

Splitting Fields: Examples

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✓ 8 results · all verified · 2 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. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The base field is the splitting field of the empty family and of every nonzero constant polynomial

Example

The splitting field of x²-2 over ℚ is ℚ(√2), with roots ±√2

Example

The splitting field of x³-2 over ℚ is ℚ(³√2,ω) with ω=(-1+i√3)/2

Example

The splitting field of x⁴+2x²-8 over ℚ is ℚ(√2,i)

Example

The splitting field of {x²-2,x²-3} over ℚ is ℚ(√2,√3)

Example

Over F₂, x⁴+x²+1=(x²+x+1)² has two distinct roots, each repeated, in its four-element splitting field

Counterexample

Adjoining one root need not split the polynomial: ℚ(³√2) does not split x³-2

False statement

FALSE: the isomorphism between two splitting fields that fixes the base field is unique

Sources