Consider the following problem: Given a prime and an integer , find an irreducible polynomial of degree . And do so in time polynomial in and . There is a randomized algorithm which attacks this problem by picking a polynomial at random (approximately out of every polynomials will be irreducible), and testing each for irreducibility (which is fast), continuing this procedure until you find one, then stop. But we are interested here in a deterministic algorithm. Already for , this is a difficult problem, equivalent to finding a quadratic nonresidue modulo .
Assuming the ERH, Adleman and Lenstra have a solution to this problem. Unconditionally, they also find an irreducible polynomial of degree with , where is an effectively computable number. Letting and (where you do not know a priori if is prime), the polynomial produced has degree , and the runtime of the AKS algorithm becomes . We improve this theorem to the following:
Theorem. Such a polynomial can be produced with for sufficiently large and .
The bound for the `sufficiently large' part depends effectively on the choice of .
Theorem. There is an effectively computable function and a deterministic algorithm such that if , , and , the algorithm produces pairs such that for each , is prime, , , the order of modulo is . Further, the are pairwise coprime, and . This algorithm runs in time .
Let be the Gaussian period of degree in (the trace of into the unique subfield of degree over ). The element has degree over the rationals (by coprimality). If is prime, and if is the minimal polynomial of then is irreducible over . Checking if in (together with some other conditions), we see that gives rise to a pseudofield.
Let . Throughout, we assume that is `sufficiently large' with the bound being effectively computable, depending only on the choice of .
Proposition. All but primes have a prime with and the order of modulo is equal to .
Therefore up to , almost all of the primes are useful in the context of our theorem. This proposition is a natural extension of the argument in the original AKS paper, together with an added ingredient about the distribution of primes such that is smooth due to Pomerance and Shperlinski.
Proposition. Let be a set of primes with and . Then there are primes such that is free of primes from .
This proposition follows from a method of Balog, together with some effective estimates on the distribution of primes in residue classes. Together, these two propositions give the corollary:
Corollary. Let be the set of primes in the first proposition satisfying . Then
Proposition. There exists a subset of in the corollary with product in the interval .
The proof of this theorem relies upon combinatorial number theory (essentially, you can solve a bin packing problem using the primes ). It relies upon:
Theorem. [Continuous Frobenius theorem] If is an open subset of , is closed under addition, and , then for any , , the measure of is , i.e.
Now a remark about effectivity. It was proven by de la Vallée Poussin in 1896 that
Now we give an outline of the proof of the Frobenius theorem. First, it is sufficient to prove the case where (i.e. contains only finitely many intervals). Second, since , for all either or (*). Now fix such that and consider all sets satisfying (**) as well as condition (*). Under these conditions, there exists a maximum to . We may thus assume that is a maximum. We show in the paper that we can assume that .
Let with . Let . For all and ,
Let such that for all . Then there exists such that for all ,
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