Kevin J. McGown

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math.NT2011

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Kevin J. McGown

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $Δ=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,…

math.NT2010

On the second smallest prime non-residue

Kevin J. McGown

Let $χ$ be a non-principal Dirichlet character modulo a prime $p$ . Let $q_{1} < q_{2}$ denote the two smallest prime non-residues of $χ$ . We give explicit upper bounds on $q_{2}$ that impr…

math.NT2010

On the constant in Burgess' bound for the number of consecutive residues or non-residues

Kevin J. McGown

We give an explicit version of a result due to D. Burgess. Let $χ$ be a non-principal Dirichlet character modulo a prime $p$ . We show that the maximum number of consecutive integer…

math.NT2010

Norm-Euclidean Galois fields

Kevin J. McGown

Let K be a Galois number field of prime degree $ℓ$ . Heilbronn showed that for a given $ℓ$ there are only finitely many such fields that are norm-Euclidean. In the case of $\e…