P-adic valuationIn number theory, the p-adic valuation or p-adic order of an integern is the exponent of the highest power of the prime numberp that dividesn. It is denoted νp(n){\displaystyle u _{p}(n)}. Equivalently, νp(n){\displaystyle u _{p}(n)} is the exponent to which p{\displaystyle p} appears in the prime factorization of n{\displaystyle n}.
CoefficientIn mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as a, b and c).
MultisetIn mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset.PolynomialType of mathematical expressionIn mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x{\displaystyle x} is x2−4x+7{\displaystyle x^{2}-4x+7}.
Integer factorizationUnsolved problem in computer scienceCan integer factorization be solved in polynomial time on a classical computer?More unsolved problems in computer scienceIn mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number.
Formal derivativeIn mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general impossible to define for a ring.Fundamental theorem of algebraThe fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.IntegerAn integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold Z {\displaystyle \mathbb {Z} } .
Field (mathematics)Algebraic structure with addition, multiplication, and divisionA field is an algebraic structure that is closed under the four usual arithmetic operations (except when it comes to division by zero).In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined.
Zero of a functionIn mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued functionf{\displaystyle f}, is a member x{\displaystyle x} of the domain of f{\displaystyle f} such that f(x){\displaystyle f(x)}vanishes at x{\displaystyle x}; that is, the function f{\displaystyle f} attains the value of 0 at x{\displaystyle x}, or equivalently, x{\displaystyle x} is a solution to the equation f(x)=0{\displaystyle f(x)=0}.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
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