characteristic

adj

1. Being a distinguishing feature of a person or thing.

   “All this was extraordinarily distasteful to Churchill. It was ugly, gross. Never before had he felt such repulsion when the vicar displayed his characteristic bluntness or coarseness of speech. In the present connexion […] such talk had been distressingly out of place.”

noun

1. A distinguishing feature of a person or thing, a part of mental or physical behavior.
2. The integer part of a logarithm.

   “It is evident, moreover, that as the logarithms of numbers, which are tenfold, the one of the other, do not differ except in their characteristics, it is sufficient that the tables contain the fractional parts only of the logarithms.”

   “As the sine and cosine are always proper fractions their logarithms are negative, i.e. have negative characteristics. When we are given an angle, it is impossible to say, from inspection of the angle, what the characteristic of the logarithm of its sine, cosine or tangent may be; so the characteristics have to be printed with the mantissae.”

   “Similarly, the characteristic for .003 is −3, and the characteristic for .0003 is −4.”

3. The distinguishing features of a navigational light on a lighthouse etc by which it can be identified (colour, pattern of flashes etc.).
4. For a given field or ring, a natural number that is either the smallest positive number n such that n instances of the multiplicative identity (1) summed together yield the additive identity (0) or, if no such number exists, the number 0.

   “The characteristic of a field, if non-zero, must be a prime number.”

   “1962 [John Wiley & Sons], Nathan Jacobson, Lie Algebras, 1979, Dover, page 289, In this chapter we study the problem of classifying the finite-dimensional simple Lie algebras over an arbitrary field of characteristic 0.”

   “1992, Simeon Ivanov (translator), P. M. Gudivok, E. Ya. Pogorilyak, On Modular Representations of Finite Groups over Integral Domains, Simeon Ivanov (editor), Galois Theory, Rings, Algebraic Groups and Their Applications, American Mathematical Society, page 87, Let R be a Noetherian factorial ring of characteristic p which is not a field.”

   “Traditionally, a complete, discretely valued field of characteristic zero, the maximal ideal of whose valuation ring is generated by the prime number p, has been called a p-adic field. In our terminology, the valuation ring of a p-adic field is a Cohen ring of characteristic zero whose residue field has characteristic p, and consequently a p-adic field is simply the quotient field of such a Cohen ring.”