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3, are given. 5, it is shown how to construct irreducible polynomials of degree n = vt from irreducible polynomials of degrees v and t with gcd( v, t) 1. In the final section, a systematic way of constructing an irreducible polynomial of any given degree over a given finite field is described. + = An irreducible polynomial /(x) E Fq[x] of degree n is said to be a 39 CHAPTER 3. IRREDUCIBLE POLYNOMIALS 40 primitive polynomial if the roots of f( z) are primitive elements in Fqn . We shall not consider primitive polynomials in this chapter, but refer the reader to [17, 34].
4. FACTORJNG where the hi(z), 1 ::; i::; t, are distinct irreducible polynomials of Fq[z]. Let n = Fq[zJI(f(z)) be the ring of polynomials modulo /(z). By the Chinese remainder theorem for polynomials, there exist unique polynomials e, (z) of degree less than n, 1 ::; i ::; t, such that ei(z) = 0 (mod hj(z)), j ei(z) _ 1 (mod hi(z)). 3) ai(z ) hi(Z ) for some polynomial ai(z) E Fq[z], with gcd(ai(z), hi(z)) = 1. 4) ei(z) = 1 + bi ( Z ) hi(z ) for some b,(z) E Fq [z]. , ei(z)ej(z) == 0 (mod /(x)) if i i= j.
At, is the primitive idempotent decomposition then we need to prove that for any two distinct coordinate positions k and 1 we can find an i such that a~i) :I a~'). Let Oi be the natural homomorphism from R: to n(i) defined by Oi(b(z)) = b(z) (mod hi(z)). 10) we have that d-t i) Oi(H(y)) = II(y - Oi(zF = hi(y) . • =0 Selecting coordinate positions k and 1 is equivalent to looking at factors hdz) and h,(z). Since H(y) == hi(y) (mod hi(z)) and since hk(z) :I h, (z) we have that for some i, the coefficients of yi in H (y) (mod hk(z )) and H(y) (mod h,(z)) will be different.