I happened across the recent arXiv paper Transfinite Recursion In Higher Reverse Mathematics, and the introduction begins:
The question "What role do incomputable sets play in mathematics?" has been a central theme in modern logic for almost as long as modern logic has existed. Six years before Alan Turing formalized the notion of computability, van der Waerden
[vdW30]showed that the splitting set of a field is not uniformly computable from the field; put another way, van der Waerden demonstrated the necessity of certain incomputable sets for Galois theory.
The article referred to is
[vdW30]Bartel L. van der Waerden. Eine Bemerkung über die Unzerlegbarkeit von Polynomen. Math. Ann., 102(1):738–739, 1930
I'm afraid I don't know any German, nor am I familiar with even the basics of complexity theory. But I was surprised and interested by this connection to Galois theory, which is something much closer to mathematical home for me.
I would greatly appreciate an overview / explanation of van der Waerden's result - what does it mean to say that Galois theory needs "certain incomputable sets", and how is this result proven?