The Theory of Modules and Number Theory

[Math Amateur]

I have recently been doing some reading (skimming really) some books on number theory, particularly algebraic number theory.

While number theory seems to draw heavily on rings and fields (especially some special types of rings like Euclidean rings and domains, unique factorization domains etc), it only seems to draw very lightly on module theory ... is my impression correct?

If my impression above is correct, then why is this so ... is it to do with the history and development of number theory and module theory ... or something more fundamental ...

Hope someone can help clarify the above ...

Peter

 

[Euge]

I have recently been doing some reading (skimming really) some books on number theory, particularly algebraic number theory.

While number theory seems to draw heavily on rings and fields (especially some special types of rings like Euclidean rings and domains, unique factorization domains etc), it only seems to draw very lightly on module theory ... is my impression correct?

If my impression above is correct, then why is this so ... is it to do with the history and development of number theory and module theory ... or something more fundamental ...

Hope someone can help clarify the above ...

Peter

I think modules are used extensively in algebraic number theory, especially Galois modules and representations. Instead of looking at modules over rings (as you have been doing so far), look at modules over groups. If $G$ is a group, then a (left) $G$-module is an abelian group $M$ together with a (left) $G$-action on $M$ which satisfies $g \cdot (x + y) = g\cdot x + g\cdot y$ for all $g\in G$ and $x, y \in M$. You can view a $G$-module as a module over a ring by considering it as a $\Bbb Z[G]$-module.

If $M$ is a $G$-module such that $G$ is the Galois group of a field extension, then $M$ is called a Galois module or Galois representation. Certainly a field is a Galois module. The ring of integers $\Bbb O_K$ of an algebraic number field $K$ is a much more nontrivial example of a Galois representation.

Here is a paper which shows extensive applicability of Galois representations to not only number theory, but algebraic geometry as well.

http://www.math.ias.edu/~rtaylor/longicm02.pdf

This is at the research level, so don't focus on understanding it -- it's just an illustration.

 

[Math Amateur]

I think modules are used extensively in algebraic number theory, especially Galois modules and representations. Instead of looking at modules over rings (as you have been doing so far), look at modules over groups. If $G$ is a group, then a (left) $G$-module is an abelian group $M$ together with a (left) $G$-action on $M$ which satisfies $g \cdot (x + y) = g\cdot x + g\cdot y$ for all $g\in G$ and $x, y \in M$. You can view a $G$-module as a module over a ring by considering it as a $\Bbb Z[G]$-module.

If $M$ is a $G$-module such that $G$ is the Galois group of a field extension, then $M$ is called a Galois module or Galois representation. Certainly a field is a Galois module. The ring of integers $\Bbb O_K$ of an algebraic number field $K$ is a much more nontrivial example of a Galois representation.

Here is a paper which shows extensive applicability of Galois representations to not only number theory, but algebraic geometry as well.

http://www.math.ias.edu/~rtaylor/longicm02.pdf

This is at the research level, so don't focus on understanding it -- it's just an illustration.

Thanks so much for your help Euge ... The idea of G-modules and Galois modules sounds really interesting ... at the very least I will look up these ideas ...

Thanks again,

Peter

 

[mathbalarka]

(The theory of Galois modules is, eh, not quite research level. Patrick Morandi's excellent book on Galois theory talks about those and their relevance in Galois cohomology : http://d.violet.vn/uploads/resources/559/208082/0387947531 - Field and Galois.pdf)

then $M$ is called a Galois module or Galois representation.

Oh? I thought (standard) Galois representations were group homomorphism $\mathsf{Gal}(\Bbb {\overline Q}/\Bbb Q) \to GL_n(V)$? Never heard a module being called a Galois representation.

 

[Euge]

The theory of Galois modules is, eh, not quite research level.

I was referring to R. Taylor's paper, not the theory. It wouldn't make sense to say that a theory is or is not 'research level'. :)

 

[mathbalarka]

Ah, I see. Thanks for clarifying.

 

[Math Amateur]

Ah, I see. Thanks for clarifying.

Mathbalarka, Euge

Thanks for the posts ... interesting ...

Peter