Lagrange's theorem (number theory)

In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime. More precisely, it states that if p is a prime number and ${\displaystyle \textstyle f(x)\in \mathbb {Z} [x]}$ is a polynomial with integer coefficients, then either:

Solutions are "incongruent" if they do not differ by a multiple of p. If the modulus is not prime, then it is possible for there to be more than deg f(x) solutions.

The two key ideas are the following. Let ${\displaystyle \textstyle g(x)\in (\mathbb {Z} /p)[x]}$ be the polynomial obtained from ${\displaystyle f(x)}$ by taking the coefficients ${\displaystyle \mod p}$. Now (i) ${\displaystyle f(k)}$ is divisible by ${\displaystyle p}$ if and only if ${\displaystyle g(k)=0}$; (ii) ${\displaystyle g(k)}$ has no more roots than its degree.

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