2022/12/09 by Hou, Xiang-dong
#05E05 #11C08 #11T55 #12-08 #20B30 #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2212.04978
An irreducible polynomial over \Bbb Fq is said to be normal over \Bbb Fq if its roots are linearly independent over \Bbb Fq. We show that there is a polynomial hn(X1,…,Xn)∈\Bbb Z[X1,…,Xn], independent of q, such that if an irreducible polynomial f=Xn+a1Xn-1+⋯+an∈\Bbb Fq[X] is such that hn(a1,…,an)≠ 0, then f is normal over \Bbb Fq. The polynomial hn(X1,…,Xn) is computed explicitly for n≤ 5 and partially for n=6. When char \Bbb Fq=p, we also show that there is a polynomial hp,n(X1,…,Xn)∈\Bbb Fp[X1,…,Xn], depending on p, which is simpler than hn but has the same property. These results remain valid for monic separable irreducible polynomials over an arbitrary field with a cyclic Galois group.