Irreducibility and r-th root finding over finite fields

Constructing r-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree r^e (where r is a prime) over a given finite field 𝔽_q of characteristic p (equivalently, constructing the bigger field 𝔽_q^r^e). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its discriminant is a quadratic nonresidue in the field. We give an extension of Stickelberger's Lemma; we construct r-th nonresidues from a polynomial f for which there is a d, such that, r|d and r∤#(irreducible factor of f(x) of degree d). Our theorem has the following interesting consequences: (1) we can construct 𝔽_q^m in deterministic poly(deg(f),mlog q)-time if m is an r-power and f is known; (2) we can find r-th roots in 𝔽_p^m in deterministic poly(mlog p)-time if r is constant and r|(m,p-1). We also discuss a conjecture significantly weaker than the Generalized Riemann hypothesis to get a deterministic poly-time algorithm for r-th root finding.