# Tagged Questions

**11**

votes

**3**answers

240 views

### Origin of number theoretic invariants associated to hyperbolic 3-manifolds

I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and ...

**4**

votes

**1**answer

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, ...

**13**

votes

**2**answers

672 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

**1**answer

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 ...

**7**

votes

**4**answers

940 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

**2**answers

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. Gaussâ€™s Disquisitiones Arithmeticae, Patterson says
"Hecke [proved] a beautiful theorem on the different ...