TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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

Statement

Let $D$ be an integral domain. A nonzero polynomial $f \in D [x]$ of degree $n$ has at most $n$ distinct roots in $D$ .

Facts & Assumptions

Given: An integral domain $D$ and a nonzero polynomial $f \in D [x]$ of degree $n$ .

[L1]

If $a$ is a root of $f$ , then $f = (x - a) q$ for some polynomial $q$ (Factor theorem over a commutative ring).

[L2]

Degrees add when nonzero polynomials over a domain are multiplied (Over an integral domain, degrees add under multiplication of nonzero polynomials).

[L3]

In an integral domain, a product is zero only if one factor is zero (Zero divisor, and integral domain: a commutative ring with $1 \neq 0$ and no zero divisors).

[L4]

If a property holds at $0$ and passes from $n$ to $n + 1$ , it holds for every natural number (The principle of mathematical induction).

Proof

technique · induction

1.1basegiven

If $n = 0$ , then $f$ is a nonzero constant and has no root, proving the base case.

2.1step 1.1ihL1L2choose

For $n > 0$ , if $f$ has no root the claim is immediate; otherwise choose a root $a$ , use [L1] to write $f = (x - a) q$ , and use [L2] to obtain $\deg q = n - 1$ .

3.1step 2.1ihL3L4discharge-induction∎

If $b \neq a$ is another root, then $0 = f (b) = (b - a) q (b)$ , and [L3] gives $q (b) = 0$ because $b - a \neq 0$ ; the induction hypothesis bounds the roots other than $a$ by $n - 1$ , so $f$ has at most $n$ roots, and [L4] completes the induction.

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

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