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<title> Algebraic Number Theory </title>
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<meta name="KeyWords" content=“number field, integers, class group, lattice, unit group, zeta function>
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<h2>Università di Roma “Tor Vergata”</h2>
<h2>University of Nanjing</h2><br>
<h1><big>Algebraic number theory</big></h1>
<h2> March 10, 2026 to May 1, 2026</h2>
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Docente: Prof. René Schoof
<h2> Program </h2>
<ul>
<li> This is a course in algebraic number theory. The main topic of study are <i>number fields</i>, i.e. finite extensions of <b>Q</b> and their rings of integers. We prove the two basic finiteness theorems in this theory: the ideal class group of the ring of integers of a number field is finite and the unit group of the ring of integers of a number field is finitely generated.
<li> These two theorems are key ingredients of the proofs of some of the most important results in 20th century number theory. They play for instance fundamental roles in the proof of the famous Mordell-Weil theorem for abelian varieties and of Faltings’ proof of the Mordell conjecture.
<li> Prerequisites are basic Linear algebra and topology, basic algebra: groups, rings and fields.
</ul>
<h2>Topics</h2>
<ul>
<li> Number fields <a href="https://mathworld.wolfram.com/NumberField.html">Wolfram</a>.
<li> Rings of integers <a href="https://www.numberanalytics.com/blog/ring-of-integers-algebraic-number-theory">AI</a>.
<li> <a href="https://en-academic.com/dic.nsf/enwiki/1424406">Discriminants</a>.
<li> <a href="https://plato.stanford.edu/entries/dedekind-foundations">Dedekind</a> <a href="https://artofproblemsolving.com/wiki/index.php/Dedekind_domain">Domains</a>
<li> <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Riemann">Riemann</a>
<a href="https://www.3blue1brown.com/lessons/zeta">zeta function</a>.
<li> Dedekind <a href="https://en.wikipedia.org/wiki/Dedekind_zeta_function">
zeta function.</a>
<li> The class group in <a href="https://en-academic.com/dic.nsf/enwiki/103804">Academic</a>.
<li> <a href="https://www.math.toronto.edu/swastik/courses/rutgers/ANT-F14/lec5.pdf">Lattices</a>.
<li> <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Minkowski">Minkowski</a>'s <a href="https://www.math.cmu.edu/~ttkocz/teaching/1819/read-sem-notes.pdf">theorem</a>.
<li> <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Poisson">Poisson</a> summation
<a href="https://zhuanlan.zhihu.com/p/107540842">formula</a>.
<li> Class number <a href="https://planetmath.org/ClassNumberFormula"> formula</a>.
<li> <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/">Fourier</>
<a href=“https://www.geeksforgeeks.org/maths/fourier-transform/“>transform</a>.
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<h2>Diario delle lezioni</h2>
<ul><li>
<a href="https://www.mat.uniroma2.it/~geo2/Algebra3-2024.html">Diario</a>.
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<h2>Notes</h2>
<ul>
<li> <a href="2.Number fields.pdf">Chapter 2.</a>
<li> <a href="3.Rings of integers.pdf">Chapter 3.</a>
<li> <a href="4.Dedekind domains.pdf">Chapter 4.</a>
<li> <a href="5. Ideals of rings of integers.pdf">Chapter 5.</a>
<li> <a href="6.Finitely generated abelian groups and lattices.pdf">Chapter 6.</a>
<li> <a href="7.Ramification and discriminants.pdf">Chapter 7.</a>
<li> <a href="8.Minkowski.pdf">Chapter 8.</a>
<li> <a href="9.Dirichlet.pdf">Chapter 9.</a>
<li> An <a href="egnew2018.pdf">example</a>.
<li>
</ul>
<h2>Various</h2>
<ul> <li>
<a href="https://pari.math.u-bordeaux.fr">PARI/GP</a>.
<li> <a href="https://www.lmfdb.org/NumberField">LMFDB</a>.
<li> Lattice based <a href="https://en.wikipedia.org/wiki/Lattice-based_cryptography">cryptography</a>.
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<h2>Material</h2>
<ul>
<li> <a href="http://owpdb.mfo.de/detail?photo_id=9239">Fröhlich A.</a>
and <a href="http://www.merton.ox.ac.uk/fellows_and_research/taylor.shtml">Taylor, M.</a>:
<b> Algebraic number theory, </b> Cambridge
University Press, Cambridge 1991.
<li> Marcus, D.: <b> Number fields, </b> 3rd Ed, Springer-Verlag 1977.
<li> <a href="http://www.jmilne.org/math/Personal/index.html">Milne, J.</a>: <b>Algebraic Number Theory, </b> Lecture Notes 2009
(<a href="http://www.jmilne.org/math/CourseNotes/ANT.pdf">pdf</a>).
<li> Conrad, K: <a href="https://kconrad.math.uconn.edu/blurbs">Blurbs</a>.
<li> Ono, T.: <b> An introduction to algebraic number theory, </b> Plenum Press,
New York 1990.
<li> Samuel, P.: <b> Théorie algébrique des nombres, </b> Hermann,
Paris 1971.
<li> Bianchi, L.: <a href="bianchi1923.pdf">Lezioni</a>
<b> sulla teoria dei numeri algebrici, </b> Pisa 1923.
<li>Schoof, R.: <b> Algebraic Number Theory, </b> under construction.
</ul>
<h2>Extra</h2>
<ul>
<li> Il <a href="https://en.wikipedia.org/wiki/Stark-Heegner_theorem">
teorema</a> di Heegner-Baker-Stark.
<li> <a href="https://math.stackexchange.com/questions/4679993/why-exactly-is-e-pi-sqrt163-a-near-integer">Why</a> <it>is</it><sup>π√163</sup> very close to an integer?
<li> <a href="riemann1859.pdf">Manuscript</a> <a href="http://en.wikipedia.org/wiki/Bernhard_Riemann">
Riemann</a> (<a href="Wilkins_D.pdf">D</a>), (<a href="Wilkins_E.pdf">E</a>).
<li> <a href="https://www.youtube.com/watch?v=sD0NjbwqlYw">Video</a>: analytic continuation and the zeta function.
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