Integral domain in the context of "Valuation ring"

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the ring to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as ${\displaystyle \left(x-{\sqrt {2}}\right)\left(x+{\sqrt {2}}\right)}$ if it is considered as a polynomial with real coefficients. One says that the polynomial x − 2 is irreducible over the integers but not over the reals.

Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common definitions. Most often, a polynomial over an integral domain R is said to be irreducible if it is not the product of two polynomials that have their coefficients in R, and that are not unit in R. Equivalently, for this definition, an irreducible polynomial is an irreducible element in a ring of polynomials over R. If R is a field, the two definitions of irreducibility are equivalent. For the second definition, a polynomial is irreducible if it cannot be factored into polynomials with coefficients in the same domain that both have a positive degree. Equivalently, a polynomial is irreducible if it is irreducible over the field of fractions of the integral domain. For example, the polynomial ${\displaystyle 2(x^{2}-2)\in \mathbb {Z} [x]}$ is irreducible for the second definition, and not for the first one. On the other hand, ${\displaystyle x^{2}-2}$ is irreducible in ${\displaystyle \mathbb {Z} [x]}$ for the two definitions, while it is reducible in ${\displaystyle \mathbb {R} [x].}$

In mathematics, negative two or minus two is an integer two units from the origin, denoted as −2 or 2. It is the additive inverse of 2, following −3 and preceding −1, and is the largest negative even integer. Except in rare cases exploring integral ring prime elements, negative two is generally not considered a prime number.

Negative two is sometimes used to denote the square reciprocal in the notation of SI base units, such as m·s. Additionally, in fields like software design, −1 is often used as an invalid return value for functions, and similarly, negative two may indicate other invalid conditions beyond negative one. For example, in the On-Line Encyclopedia of Integer Sequences, negative one denotes non-existence, while negative two indicates an infinite solution.

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.

The field of fractions of an integral domain ${\displaystyle R}$ is sometimes denoted by ${\displaystyle \operatorname {Frac} (R)}$ or ${\displaystyle \operatorname {Quot} (R)}$, and the construction is sometimes also called the fraction field, field of quotients, or quotient field of ${\displaystyle R}$. All four are in common usage, but are not to be confused with the quotient of a ring by an ideal, which is a quite different concept. For a commutative ring that is not an integral domain, the analogous construction is called the localization or ring of quotients.

In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.