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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.
every element x of GF(2) satisfies x + x = 0 and therefore −x = x; this means that the characteristic of GF(2) is 2; every element x of GF(2) satisfies x 2 = x (i.e. is idempotent with respect to multiplication); this is an instance of Fermat's little theorem. GF(2) is the only field with this property (Proof: if x 2 = x, then either x = 0 or ...
The monic irreducible polynomial x 8 + x 4 + x 3 + x 2 + 1 over GF(2) is primitive, and all 8 roots are generators of GF(2 8). All GF(2 8 ) have a total of 128 generators (see Number of primitive elements ), and for a primitive polynomial, 8 of them are roots of the reducing polynomial.
This implies that, over GF(2), there are exactly 9 = 54 / 6 irreducible monic polynomials of degree 6. This may be verified by factoring X 64 − X over GF(2) . The elements of GF(64) are primitive n {\displaystyle n} th roots of unity for some n {\displaystyle n} dividing 63 {\displaystyle 63} .
Irreducible polynomials over finite fields are also useful for pseudorandom number generators using feedback shift registers and discrete logarithm over F 2 n. 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 ...
Over GF(2), x + 1 is a primitive polynomial and all other primitive polynomials have an odd number of terms, since any polynomial mod 2 with an even number of terms is divisible by x + 1 (it has 1 as a root). An irreducible polynomial F(x) of degree m over GF(p), where p is prime, is a primitive polynomial if the smallest positive integer n ...
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 ...
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]).