An essay I wrote on counting number fields during Cambridge Part III.

A response to the question "What's a mathematician to do?" by Bill Thurston.

Some of my thoughts on consensus in mathematics. Some of my beliefs are likely to have changed since writing.

Doing mathematics sounds a lot like hunting for four-leaf clovers.

A visual story about the insolubility of the quintic (by the YouTube channel 2swap) following Vladimir Arnold's proof.

A visual primer on braid groups and configuration spaces (by Chris Staecker).

Does there exist a number field K such that there is a finite extension L/K that ramifies above a real archimedean place of K and nowhere else?

According to GPT-5.5 Pro, the answer is yes (as predicted by analogy with function fields or 3-manifolds). The first example of K it found was Example 2 in a paper of Dummit and Kisilevsky (which it gave credit to). An "optimized" example based on the ideas of Dummit–Kisilevsky is the following. Consider K = Q(√231), H = Q(√3, √77), ∞ the positive embedding of K, T the set of places of H above ∞, and L the ray class field of H of modulus T.

In hindsight, it is not surprising that the ideas of Dummit–Kisilevsky proved helpful in answering this question, although executing a similar literature search on my own would have taken significantly more time than the 35 minutes and 40 seconds taken by GPT. If you have a "simpler" example than the field K = Q(√231), I would be curious to know.