It is helpful to compare irreducible polynomials to prime numbers : prime numbers ( together with the corresponding negative numbers of equal magnitude ) are the irreducible integers.
22.
Since the substitution is an automorphism of the ring, the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally.
23.
Since the substitution is an automorphism of the ring, the fact that we obtain an irreducible polynomial after substitution implies that we had an irreducible polynomial originally.
24.
One of these first results was irreducible polynomial " f " with integer coefficients such that " f " ( ? ) = 0.
25.
If denotes the finite field of order ( where is necessarily a prime power ), then the number of monic irreducible polynomials of degree over is given by:
26.
To ensure that this field is actually-dimensional and does not collapse to an even smaller field, it is sufficient that is an irreducible polynomial over the rationals.
27.
Thus for any irreducible polynomial whose degree is not a power of 2 and which has all roots real, the roots cannot be expressed purely in terms of real radicals.
28.
They exhibit many of the general properties of the concept of " irreducibility " that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors.
29.
This implies that " p " is an irreducible polynomial, and thus that the quotient ring K [ X ] / \ langle p \ rangle is a field.
30.
For p = 2 and p = 3, it's not difficult to get a desired factorization into irreducible polynomials ( which can be prime, by the hypothesis ).
