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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.
The converse of this criterion is that, if p is an irreducible polynomial with integer coefficients that have greatest common divisor 1, then there exists a base such that the coefficients of p form the representation of a prime number in that base.
To see this, choose a monic irreducible polynomial f(X 1, ..., X n, Y) whose root generates N over E. If f(a 1, ..., a n, Y) is irreducible for some a i, then a root of it will generate the asserted N 0.) Construction of elliptic curves with large rank. [2] Hilbert's irreducibility theorem is used as a step in the Andrew Wiles proof of Fermat's ...
In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial. In representation theory, an irreducible representation is a nontrivial representation with no nontrivial proper subrepresentations. Similarly, an irreducible module is another name for a simple module.
For a concrete example one can take R = Z[i√5], p = 1 + i√5, a = 1 − i√5, q = 2, b = 3. In this example the polynomial 3 + 2X + 2X 2 (obtained by dividing the right hand side by q = 2) provides an example of the failure of the irreducibility statement (it is irreducible over R, but reducible over its field of fractions Q[i√5]).
Imperfect fields cause technical difficulties because irreducible polynomials can become reducible in the algebraic closure of the base field. For example, [ 4 ] consider f ( x , y ) = x p + a y p ∈ k [ x , y ] {\displaystyle f(x,y)=x^{p}+ay^{p}\in k[x,y]} for k {\displaystyle k} an imperfect field of characteristic p {\displaystyle p} and a ...
The number of irreducible monic polynomials of degree n over F q is the number of aperiodic necklaces, given by Moreau's necklace-counting function M q (n). The closely related necklace function N q (n) counts monic polynomials of degree n which are primary (a power of an irreducible); or alternatively irreducible polynomials of all degrees d ...
If f(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x 2 − 2 is irreducible over the rational numbers and has 2 {\displaystyle {\sqrt {2}}} as a root; hence there is no linear or constant polynomial over the rationals having 2 {\displaystyle {\sqrt {2}}} as a root.