In Samuel James Patterson's article titled Gauss Sums in The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, Patterson says
"Hecke [proved] a beautiful theorem on the different of k, namely that the class of the absolute different in the ideal class group is a square. This theorem - an analogue of the fact that the Euler characteristic of a Riemann surface is even - is the crowning moment (coronidis loco) in both Hecke's book and Andre Weil's Basic Number Theory."
About the same matter, J.V. Armitage says (in his review of the 1981 translation of Hecke's book):
"That beautiful theorem deservedly occupies the 'coronidis loco' in Weil's Basic number theory and was the starting point for the work on parity problems in algebraic number theory and algebraic geometry, which has borne such rich fruit in the past fifteen years."
What is a reference for learning about the parity problems that Armitage alludes to?
It can be impossible to verbalize the reasons for aesthetic preferences, but
Why might Weil, Patterson and Armitage have been so favorably impressed by the theorem that the ideal class of the different of a number field is a square in the ideal class group?
Weil makes no comment on why he chose to end Basic Number Theory with the above theorem. It should be borne in mind that Weil's book covers the class number formula and all of class field theory, so that the standard against which the above theorem is being measured in the above quotes is high!