Manchester Geometry Seminar 1999/2000

The talk will begin by briefly recounting some well-known results in the Galois theory and Hermitian-Galois theory of algebraic number fields. I shall then go on to discuss analogues for higher dimensional arithmetic varieties. The unifying theme is that of Euler characteristics (of varying degrees of sophistication). I hope to conclude by describing the use of Quillen metrics to produce an Arakelov-Euler characteristic.