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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.
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.
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].}
Gauss's lemma (irreducibility) — A non-constant polynomial in Z[X] is irreducible in Z[X] if and only if it is both irreducible in Q[X] and primitive in Z[X]. The proof is given below for the more general case.
Consider the polynomial Q(x) = 3x 4 + 15x 2 + 10.In order for Eisenstein's criterion to apply for a prime number p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10.
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.
That said, Vanessa Rissetto, R.D., co-founder of the virtual nutrition care service Culina Health, believes the FDA's move is a positive step forward."Early-stage cancers are on the rise across ...
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element in a commutative ring is called prime if, whenever for some and in , then or . ...