Alphabeta Math

Polynomial Rings and Roots: Examples and Counterexamples

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✓ 14 results · all verified · 0 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 14 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

Polynomial addition and multiplication computed from coefficient convolution

Example

Long division, the Euclidean algorithm and a Bézout identity in ℚ[x]

Counterexample

Over ℤ/4, a nonconstant polynomial can be a unit and product degree can drop

Counterexample

Division with a degree-small remainder can fail over ℤ when the leading coefficient of the divisor is not a unit

Counterexample

Quadratics can have four roots over ℤ/6 and ℤ/8

Counterexample

Over Fₚ, xᵖ-x and the zero polynomial induce the same function but are distinct polynomials

Example

The polynomial x²-2 is irreducible over ℚ

Example

The polynomial x²+1 is irreducible over ℝ

Example

Reduction modulo 2 proves x³+17x+391 irreducible over ℚ

Example

Eisenstein proves xⁿ-2 irreducible over ℚ for every positive n

Example

Translation turns x⁴+1 into an Eisenstein polynomial

Counterexample

A degree-four polynomial can be reducible over ℚ without having a rational root

Example

In characteristic 2, x²+1=(x+1)² has zero derivative and a repeated root

Counterexample

For a field F, the ideal (x,y) in F[x,y] is not principal

Sources