Norm-Euclidean Galois fields

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified, but for $\ell\neq 2$, little else is known. We give the first upper bounds on the discriminants of such fields when $\ell>2$. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $\Delta=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for $\zeta_K(s)$. If $38(\ell-1)^2(\log f)^6\log\log f<f$, then $K$ is not norm-Euclidean

An improved error term for counting D4D_4-quartic fields

We prove that the number of quartic fields $K$ with discriminant $|\Delta_K|\leq X$ whose Galois closure is $D_4$ equals $CX+O(X^{3/5+\epsilon})$, improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier.Comment: 11 pages, 1 figur

Computation of the least primitive root

Let $g(p)$ denote the least primitive root modulo $p$, and $h(p)$ the least primitive root modulo $p^2$. We computed $g(p)$ and $h(p)$ for all primes $p\le 10^{16}$. Here we present the results of that computation and prove three theorems as a consequence. In particular, we show that $g(p)<p^{5/8}$ for all primes $p>3$ and that $h(p)<p^{2/3}$ for all primes $p$

Counting quintic fields with genus number one

We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we compute the average genus number of quintic fields. All of these results also hold when restricted to $S_5$-quintic fields only.Comment: 7 page