Alphabeta Math

Field Extensions and the Complex Numbers: Examples and Counterexamples

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✓ 7 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

ℚ(√2)≅ℚ[x]/(x²-2) with basis 1,√2

Example

The four-element field (ℤ/2)[x]/(x²+x+1)

Counterexample

ℚ[x]/(x²-1) is not a field: reducibility creates nonzero zero divisors

Example

The minimal polynomial of √2+√3 over ℚ is x⁴-4x²+1

Example

The square roots of i are ±(1+i)/√2

False statement

FALSE: the complex field admits an order making it an ordered field

Counterexample

An annihilating polynomial need not be minimal: √2 is a root of both x²-2 and x⁴-4

Sources