Up to in the context of Algebraic closure

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

The property of being prime is called primality. A simple but slow method of checking the primality of a given number ⁠${\displaystyle n}$⁠, called trial division, tests whether ⁠${\displaystyle n}$⁠ is a multiple of any integer between 2 and ⁠${\displaystyle {\sqrt {n}}}$⁠. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of October 2024 the largest known prime number is a Mersenne prime with 41,024,320 decimal digits.

In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely as a product of prime numbers, up to the order of the factors. For example,

In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The order of the magic square is the number of integers along one side (n), and the constant sum is called the magic constant. If the array includes just the positive integers ${\displaystyle 1,2,...,n^{2}}$, the magic square is said to be normal. Many authors take magic square to mean normal magic square.

Magic squares that include repeated entries do not fall under this definition and are referred to as trivial. Some well-known examples, including the Sagrada Família magic square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square).

In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as ⁠${\displaystyle A\cong B}$⁠. The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified.

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

In mathematics, the sumset of two subsets A and B of an (additive) abelian group is formed by adding each element of A to each element of B:$${\displaystyle A+B=\{a+b\mid a\in A,\ b\in B\}.}$$

In geometry, the Minkowski sum of two subsets A and B of a Euclidean space is the set of the points whose position vectors form the sumset of the position vectors of A and B. The Minkowski sum depends on the choice of an origin in the Euclidean space. As a change of origin amounts to translate the Minkowski sum, the Minkowski sum is defined up to a translation, and its shape and orientation are well defined.

In mathematics, the term modulo ("with respect to a modulus of", the Latin ablative of modulus which itself means "a small measure") is often used to assert that two distinct mathematical objects can be regarded as equivalent—if their difference is accounted for by an additional factor. It was initially introduced into mathematics in the context of modular arithmetic by Carl Friedrich Gauss in 1801. Since then, the term has gained many meanings—some exact and some imprecise (such as equating "modulo" with "except for"). For the most part, the term often occurs in statements of the form:

which is often equivalent to "A is the same as B up to C", and means

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups ${\displaystyle A}$ and ${\displaystyle B}$ is another abelian group ${\displaystyle A\oplus B}$ consisting of the ordered pairs ${\displaystyle (a,b)}$ where ${\displaystyle a\in A}$ and ${\displaystyle b\in B}$. To add ordered pairs, the sum is defined ${\displaystyle (a,b)+(c,d)}$ to be ${\displaystyle (a+c,b+d)}$; in other words, addition is defined coordinate-wise. For example, the direct sum ${\displaystyle \mathbb {R} \oplus \mathbb {R} }$, where ${\displaystyle \mathbb {R} }$ is real coordinate space, is the Cartesian plane, ${\displaystyle \mathbb {R} ^{2}}$. A similar process can be used to form the direct sum of two vector spaces or two modules.

Direct sums can also be formed with any finite number of summands; for example, ${\displaystyle A\oplus B\oplus C}$, provided ${\displaystyle A,B,}$ and ${\displaystyle C}$ are the same kinds of algebraic structures (e.g., all abelian groups, or all vector spaces). That relies on the fact that the direct sum is associative up to isomorphism. That is, ${\displaystyle (A\oplus B)\oplus C\cong A\oplus (B\oplus C)}$ for any algebraic structures ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ of the same kind. The direct sum is also commutative up to isomorphism, i.e. ${\displaystyle A\oplus B\cong B\oplus A}$ for any algebraic structures ${\displaystyle A}$ and ${\displaystyle B}$ of the same kind.

In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property.

Technically, a universal property is defined in terms of categories and functors by means of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below).

In calculus, the constant of integration, often denoted by ${\displaystyle C}$ (or ${\displaystyle c}$), is a constant term added to an antiderivative of a function ${\displaystyle f(x)}$ to indicate that the indefinite integral of ${\displaystyle f(x)}$ (i.e., the set of all antiderivatives of ${\displaystyle f(x)}$), on a connected domain, is only defined up to an additive constant. This constant expresses an ambiguity inherent in the construction of antiderivatives.

More specifically, if a function ${\displaystyle f(x)}$ is defined on an interval, and ${\displaystyle F(x)}$ is an antiderivative of ${\displaystyle f(x),}$ then the set of all antiderivatives of ${\displaystyle f(x)}$ is given by the functions ${\displaystyle F(x)+C,}$ where ${\displaystyle C}$ is an arbitrary constant (meaning that any value of ${\displaystyle C}$ would make ${\displaystyle F(x)+C}$ a valid antiderivative). For that reason, the indefinite integral is often written as ${\textstyle \int f(x)\,dx=F(x)+C,}$ although the constant of integration might be sometimes omitted in lists of integrals for simplicity.

In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings.

Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient ring ${\displaystyle \mathbb {Z} /mn\mathbb {Z} }$ is the product of ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ and ${\displaystyle \mathbb {Z} /n\mathbb {Z} .}$