4
votes
1answer
1k views

Algebraic number theory: building and simplifying

This is a somewhat subjective question, about the past, present and especially future of algebraic number theory. I'm not at all in this area, but I'd be interested in an answer. As we all know, ...
12
votes
2answers
628 views

Did Hermite really prove “Hermite's Theorem” on number field discriminants?

Hermite's theorem, as it is typically called, is that there are only finitely many number fields of bounded (equivalently, fixed) discriminant. The usual proof (see Neukirch's Algebraic Number Theory ...
5
votes
1answer
1k views

Motivation for the proof of Hilbert's Theorem 90

The proof of Hilbert's Theorem 90 about cyclic extensions goes like this: Let $\sigma$ be the generator of the Galois group of order $n$ and let $b$ have norm $1$, i.e. $b \sigma(b) \cdots ...
6
votes
4answers
916 views

Origins of functional field arithmetic

Background: By function field, we mean a finite extension of the field of rational functions of one variable over a finite field with $p$ elements. Classfield theory for function fields was ...
15
votes
2answers
1k views

Context for “Coronidis Loco” from Weil's Basic Number Theory

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 ...