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In this expository article, we compare Malle's conjecture on counting number fields of bounded discriminant with recent conjectures of Ellenberg--Satriano--Zureick-Brown and Darda--Yasuda on counting points of bounded height on classifying stacks. We illustrate the comparisons via the classifying stacks $B(\mathbb{Z}/n\mathbb{Z})$ and $B{\mu_n}$.

Let $K$ be an algebraic number field and $H$ the absolute Weil height. Write $c_K$ for a certain positive constant that is an invariant of $K$. We consider the question: does $K$ contain an algebraic integer $\alpha$ such that both $K = \mathbb{Q}(\alpha)$ and $H(\alpha) \le c_K$? If $K$ has a real embedding then a positive answer was established in previous work. Here we obtain a positive answer if $\textrm{Tor}\bigl(K^{\times}\bigr) \not= \{\pm 1\}$, and so $K$ has only complex embeddings. We also show that if the answer is negative, then $K$ is totally complex, $K$ has a subfield $F$ that is totally real, $K/F$ is a Galois extension, and $\textrm{Tor}\bigl(K^{\times}\bigr) = \{\pm 1\}$., Comment: The statement and proof of Theorem 1.3 are corrected. We are grateful to Professor Hendrik Lenstra for pointing out an error in the previous draft

Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations., Comment: a version of this article has been accepted for publication in Essential Number Theory