A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots

Depends on

• Factor theorem over a commutative ring
• Over an integral domain, degrees add under multiplication of nonzero polynomials
• Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors
• The principle of mathematical induction

Used by

• Every finite subgroup of the unit group of an integral domain is cyclic Corollary
• On the affine line, the classical Zariski topology is cofinite Corollary
• Over an infinite integral domain, equal polynomial functions come from equal polynomials Corollary
• ∑_n≥0n!xⁿ is a formal power series that is not rational over ℚ Counterexample
• P(x)=x(x-1) vanishes on {0,1} although degₓ P=|{0,1}| Counterexample
• Quadratics can have four roots over ℤ/6 and ℤ/8 Counterexample
• Evaluation at four distinct real points gives an inner product on polynomials of degree at most three Example
• A false sum-check claim rarely becomes true Lemma
• A polynomial vanishing at every tuple from an infinite subdomain is the zero polynomial Lemma
• For a degree-n extension of a field of order q, the q-power map has order exactly n Lemma
• For prime q and d≥1, the congruence xᵈ≡1 (mod q) has at most d residue-class solutions Lemma
• The elements of a finite extension fixed by the q-power map are exactly the base field Lemma
• A field with q elements is the splitting field of x^q-x over its prime subfield Proposition
• Every F-endomorphism of a splitting field permutes the distinct roots and is an automorphism Proposition
• μₙ(K) is cyclic of order dividing n, and has a primitive n-th root of unity exactly when its order is n Proposition
• Existing polynomial identity bounds Remark
• Formal polynomials are not the functions they induce Remark
• A monic irreducible of degree d over F_q has the d distinct roots α,α^q,…,α^qᵈ⁻¹ Theorem
• Euler's criterion: (a/p)≡ a^(p-1)/2 (mod p) Theorem
• For every n≥1 there are infinitely many primes p with p≡1 (mod n) Theorem
• If deg_xᵢP<| Sᵢ| for each i and P vanishes on S₁×⋯× Sₙ, then P=0 Theorem
• Schwartz zippel over finite fields Theorem
• The Schwartz-Zippel lemma Theorem
• Φₙ is irreducible in ℚ[t] for every n≥1 Theorem

Dependency tree · two levels

14 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

• Thomas W. Judson, Abstract Algebra: Theory and Applications, Corollary 17.9 (standard reference, not scraped)
• Neil Donaldson, Math 120B Notes, Theorem 23.14 (standard reference, not scraped)