Nicola Mazzari

Nicola Mazzari

helping students in solving linear systems | struggling with conjectures in arithmetic algebraic geometry | cycling a lot

An Introduction to Number Theory

Published:

Course presentation — Padova, September 2025

What is number theory?

In the third century, Diophantus of Alexandria investigated problems of the type:
Given \(f \in \mathbb{Z}\left[X_1, ..., X_n\right]\), can we find integer (or rational) solutions to:

\[f(X_1, ..., X_n) = 0 \quad ?\]

Number theory is broadly the study of this question and its generalizations, such as systems of polynomial equations.

Examples

  1. Linear Diophantine equation:
\[aX_1 + bX_2 = m \quad \text{where } a, b, m \in \mathbb{Z}\]

This is handled by the Euclidean algorithm.

  1. Fermat’s equation:
\[X^n + Y^n = Z^n\]

which has no non-trivial integer solutions for \(n > 2\). We will prove this when \(n\) is a regular prime.

A toy example

Consider:

\[X^3 = Y^2 + 2\]

Factoring over \(\mathbb{Z}\left[ \sqrt{-2} \right]\):

\[X^3 = (Y + \sqrt{-2})(Y - \sqrt{-2})\]

Using:

  • \(\mathbb{Z}\left[ \sqrt{-2}\right]\) is a UFD
  • Its units are \(\pm1\)
  • \(Y + \sqrt{-2}\) and \(Y - \sqrt{-2}\) are coprime

We conclude the only solutions are \((X, Y) = (3, \pm5)\).

Numbers like \(Y + \sqrt{-2}\) were once called ideal numbers — inspiring the modern concept of ideals in rings.

Conclusion

Solving equations like \(aX + bY = m\) relies on unique factorization in \(\mathbb{Z}\). Similarly, solving \(X^3 = Y^2 + 2\) uses factorization in \(\mathbb{Z}\left[\sqrt{-2}\right]\). But in general, we must deal with rings that are not UFDs.

Hence, we develop the theory of number fields.

Let \(K\) be a number field (a finite extension of \(\mathbb{Q}\)), and \(\mathcal{O}_K\) its ring of integers. Then:

  1. Every ideal \(I \subset \mathcal{O}_K\) has a unique factorization:
\[I = P_1^{e_1} \cdots P_n^{e_n}\]
  1. The unit group \(\mathcal{O}_K^\times\) is a finitely generated abelian group with:
\[\text{rank}(\mathcal{O}_K^\times) = r_1 + r_2 - 1\]

where \(r_1\) = number of real embeddings, \(r_2\) = number of complex pairs.

  1. The Dedekind zeta function \(\zeta_K(s)\) satisfies:
\[\lim_{s\to 1}(s - 1)\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2} h_K R_K}{\omega_K \sqrt{|d_K|}}\]

where all the main arithmetic invariants of the field \(K\) show up.

During the course we will review all the basics of algebraic and analytic number theory, along with a lot of exercises and concrete examples.