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A relative integral basis over $ℚ\left(\sqrt{-3}\right)$ for the normal closure of a pure cubic field.

International Journal of Mathematics and Mathematical Sciences

Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics

Journal de Théorie des Nombres de Bordeaux

Explicit normal integral bases are given for some cyclic quintic fields defined by Emma Lehmer’s parametric family of quintics.

The conductor of a cyclic quartic field using Gauss sums

Czechoslovak Mathematical Journal

Let $Q$ denote the field of rational numbers. Let $K$ be a cyclic quartic extension of $Q$. It is known that there are unique integers $A$, $B$, $C$, $D$ such that $K=Q\left(\sqrt{A\left(D+B\sqrt{D}\right)}\right),$ where $A\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4.0pt}{0ex}}\text{squarefree}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}\text{odd},D={B}^{2}+{C}^{2}\phantom{\rule{4pt}{0ex}}\text{is}\phantom{\rule{4.0pt}{0ex}}\text{squarefree},\phantom{\rule{4pt}{0ex}}B>0,\phantom{\rule{4pt}{0ex}}C>0,GCD\left(A,D\right)=1.$ The conductor $f\left(K\right)$ of $K$ is $f\left(K\right)={2}^{l}|A|D$, where $l=\left\{\begin{array}{c}3,\phantom{\rule{1.0em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}D\equiv 2\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}D\equiv 1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}B\equiv 1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}2\right),\hfill \\ 2,\phantom{\rule{1.0em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}D\equiv 1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}B\equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}2\right),\phantom{\rule{4pt}{0ex}}A+B\equiv 3\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right),\hfill \\ 0,\phantom{\rule{1.0em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}D\equiv 1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right),\phantom{\rule{4pt}{0ex}}B\equiv 0\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}2\right),\phantom{\rule{4pt}{0ex}}A+B\equiv 1\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right).\hfill \end{array}\right\$ A simple proof of this formula for $f\left(K\right)$ is given, which uses the basic properties of quartic Gauss sums.

On the prime factors of non-congruent numbers

Colloquium Mathematicae

We give infinitely many new families of non-congruent numbers where the first prime factor of each number is of the form 8k+1 and the rest of the prime factors have the form 8k+3. Products of elements in each family are shown to be non-congruent.

PSL$\left(2,7\right)$ septimic fields with a power basis

Journal de Théorie des Nombres de Bordeaux

We give an infinite set of distinct monogenic septimic fields whose normal closure has Galois group $PSL\left(2,7\right)$.

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