• Corpus ID: 118452243

Norm-Euclidean Galois fields

@article{McGown2010NormEuclideanGF,
  title={Norm-Euclidean Galois fields},
  author={Kevin J. McGown},
  journal={arXiv: Number Theory},
  year={2010}
}
In this work, we study norm-Euclidean Galois number fields. In the quadratic setting, it is known that there are finitely many and they have been classified. In 1951, Heilbronn showed that for each odd prime l, there are finitely many norm-Euclidean Galois fields of degree l. Unfortunately, his proof does not provide an upper bound on the discriminant, even in the cubic case. We give, for the first time, an upper bound on the discriminant for this class of fields. Namely, for each odd prime l… 
1 Citations

Tables from this paper

References

SHOWING 1-10 OF 55 REFERENCES
Cyclotomic fields with unique factorization.
2m For a natural number w>2, we let Cm = Q(e) be the ra-th cyclotornic field, of degree (m) over the field Q of rational numbers, we let hm denote the class number of Cm, and we let/? be a prime
Counting Points on Curves over Finite Fields
We consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) ?Fqx,y,z, which are rational over a ground field Fq. More precisely, we show
Explicit bounds for primality testing and related problems
Many number-theoretic algorithms rely on a result of Ankeny, which states that if the Extended Riemann Hypothesis (ERH) is true, any nontrivial multiplicative subgroup of the integers modulo m omits
On the Euclidean algorithm in quadratic number fields
2. Previous results. In order that a field be Euclidean the class number must be 1. However, this condition is not sufficient for, as Dedekind pointed out [l j 1 , the field R( —19) has class number
Some effective cases of the Brauer-Siegel Theorem
Let k be an algebraic number field of degree n~ and discriminant D~. We let Kk denote the residue of ~(s), " " the zeta function of k, at s = 1. One version of the Brauer-Siegel Theorem is that if k
On Euclid's Algorithm in some Cyclic Cubic Fields
We now let # be the set of points P o such that M = M(P0); if sup M(Pt) < M Pit* (which happens in all cases known so far) we call this the second minimum, M2. R(0) is Euclidean if, and only if, M{P)
Quadratic class numbers and character sums
TLDR
An explicit version of Burgess' theorem valid for prime discriminants is given and, as an application, the class number of a 32-digit discriminant is computed.
The least quadratic non residue
This, the problem of the least quadratic non residue, has often been investigated. The best result is due to Vinogradov, who proved that (1) n(k) = O(k1I(2Ve) where n(k) denotes the least positive
About Euclidean rings
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean value
...
...