The Euclidean algorithm in quintic and septic cyclic fields

An Investigation Into Several Explicit Versions of Burgess' Bound

“…Unconditional bounds on the conductor f of norm-Euclidean cyclic number fields of odd prime degree 3 ≤ l < 100. These bounds improve upon [7,Proposition 2.4] by no more than a factor of 100. Remark 1.12.…”

“…2547-48], we may take C(r) = C(r; 10 40 ), where r = 4 for l = 5, 7 and r = 3 otherwise. Then, by inequality (8.1) in [7], the bound on the conductor for l > 3 is given by the smallest f for which (5.1) f ≥ 2.7D 2 (r) r (l − 1) r f 3r+1 4r (log f ) 5.…”

“…For an application, we investigate the problem of the existence of a kth power non-residue modulo p which is less than p α for several fixed α. We also provide a quick improvement to the conductor bounds for norm-Euclidean cyclic fields found in [7].…”

Euclidean algorithm in Galois Quartic Fields

“…Recently some progress has been made to show Euclidean algorithm in real quadratic fields (see [17]), in certain cyclic cubic fields (see [16]) and for cyclic fields of higher degree the reader may refer to [11], [12].…”

Explicit bounds for small prime nonresidues

“…As we alluded to a moment ago, Banks and Guo have recently shown that q k ≪ p 1 4 √ e exp( e −1 log p log log p) in the case where χ is the Legendre symbol (see [1]), provided k ≤ p 1 8 √ e exp( 1 2 e −1 log p log log p − 1 2 log log p). Often in applications (see, for example, [7,5,10,2]) one requires estimates that are completely explicit, and one is willing to accept a weaker asymptotic in order to obtain constants of a reasonable magnitude. Our goal here is to give an explicit upper bound on q k , the kth smallest prime nonresidue.…”