Multiplicity (mathematics)

The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, double roots counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". In prime factorization, the multiplicity of a prime factor is its p {\displaystyle p} -adic valuation. For example, the prime factorization of the integer 60 is the multiplicity of the prime factor 2 is 2, while the multiplicity of each of the prime factors 3 and 5 is 1. Thus, 60 has four prime factors allowing for multiplicities, but only three distinct prime factors. Let F {\displaystyle F} be a field and p ( x ) {\displaystyle p(x)} be a polynomial in one variable with coefficients in F {\displaystyle F} .