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In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's Last Theorem. If a polynomial g ( x ) ∈ Z [ x ] {\displaystyle g(x)\in \mathbb {Z} [x]} is a perfect square for all large integer values of x , then g(x) is the square of a polynomial in Z [ x ] . {\displaystyle \mathbb {Z} [x].}
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.
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component of an algebraic set is an algebraic subset that is irreducible and maximal (for set inclusion) for this property.
The twisted cubic is a projective algebraic variety.. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers.
This equivalence does not hold for general commutative rings, which is why the assumption of the ring having no nonzero zero divisors is commonly made in the definition of irreducible elements. It results also that there are several ways to extend the definition of an irreducible element to an arbitrary commutative ring. [1]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Absolute irreducibility more generally holds over any field not of characteristic two. In characteristic two, the equation is equivalent to (x + y −1) 2 = 0. Hence it defines the double line x + y =1, which is a non-reduced scheme. The algebraic variety given by the equation + = is not absolutely irreducible.