13
votes
6answers
1k views

Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...
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 ...
3
votes
2answers
458 views

Local methods in algebraic number theory

I'm currently reading about local and global fields in number theory. I have trouble seeing the point or exactly how they help answer questions about e.g. number fields. To be more specific: What ...
1
vote
1answer
702 views

Good Minkowski Theory and Commutative Algebra Books

I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory. However, I am interested in learning algebraic number theory and I recently found that the ...
5
votes
1answer
546 views

Why are absolute values more natural than discrete valuations?

It is true that considering the archimedean places as well is more general, but that still doesn't explain why it is more natural. If we consider both the definitions of an absolute value and that of ...
6
votes
1answer
444 views

Can algebraic number fields be generalized in a similar way to function fields in 1 variable over a finite field?

Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are ...