A note on asymptotically good extensions in which infinitely many primes split completely

整数多项式在素数阶域上的分裂

Splitting of integer polynomials over fields of prime order

本文利用初等域论与数论，证明了存在无穷多个素数使得给定整数多项式恰好完全分裂为线性因子，并揭示了该分裂与特定分裂域多项式有根的等价关系。

Hyperbolic Heron Triangles and Elliptic Curves

We define hyperbolic Heron triangles (hyperbolic triangles with "rational" side-lengths and area) and parametrize them in two ways as rational points of certain elliptic curves. We show that there are infinitely many hyperbolic Heron triangles with one angle $α$ and area $A$ for any (admissible) choice of $α$ and $A$; in particular, the congruent number problem has always infinitely many solutions in the hyperbolic setting. We also explore the question of hyperbolic triangles with a rational median and a rational area bisector (median splitting the triangle in half).