teorema de Lagrange

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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 p. More precisely, it states that for all integer polynomials f ∈ Z [ x ] {\displaystyle \textstyle f\in \mathbb {Z} [x]} , either: every coefficient of f is divisible by p, or p ∣ f ( x ) {\displaystyle p\mid f(x)} has at most deg f solutions in {1, 2, ..., p}, where deg f is the degree of f. This can be stated with congruence classes as follows: for all polynomials f ∈ ( Z / p Z ) [ x ] {\displaystyle \textstyle f\in (\mathbb {Z} /p\mathbb {Z} )[x]} with p prime, either: every coefficient of f is null, or f ( x ) = 0 {\displaystyle f(x)=0} has at most deg f solutions in Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } . If p is not prime, then there can potentially be more than deg f(x) solutions. Consider for example p=8 and the polynomial f(x)=x2−1, where 1, 3, 5, 7 are all solutions.