The natural way to extend scalars on a scheme is base change, and the natural way to restrict scalars is Weil restriction. To better understand these, we can look at how they interact with each other. In particular: what happens if we take the Weil restriction of a scheme, and then base change back to the original field?
Let be a Galois extension of number fields. By the Chebotarev density theorem the proportion of primes of that are totally split in is . But from the perspective of , the situation is different: almost all primes of lie above a totally split prime of .
The Brauer-Nesbitt theorem states that a representation is uniquely determined (up to semi-simplification) by the characteristic polynomials of for all . Galois representations are often described in this way, but usually only using the characteristic polynomials of Frobenius elements, and usually away from a finite set of primes, or even on a density 1 set of primes. Let’s check that this is still enough to uniquely determine the representation.
Reduction is a useful skill to have as an arithmetic geometer. Here we examine some elliptic curves whose reductions can be described relatively easily, and at the end some curious behavior of reduction mod upon extending the base field. Nearly all of this is from the book A First Course in Modular Forms by Diamond & Shurman, and in particular exercises 8.3.6 and 8.4.4.
Another useful application of Galois theory to prime factorization in number fields is the Frobenius element associated to a prime.
Let be a Galois extension of number fields, and suppose is factorized in as with each having inertia degree . That is, is a degree extension of , which in turn is a finite extension of (where ).