Topic outline

  • General

    Welcome to the Galois Theory course!

    For the first week, my postdoc Pavlo Yatsyna will substitute me for the lectures due to travels. He is an expert in algebraic number theory, so you will be in good hands! Attendance on the first lecture is important, as we will go through the course practices there. Otherwise attendance is not mandatory, but warmly recommended :).

    As prerequisites, one should have taken the Abstract Algebra course (or otherwise possess similar knowledge). This course is less abstract, as we will (mainly) work with number fields, i.e., subfields of the complex numbers. We construct field extensions by polynomials, and study the related relative structures both from a group-theoretic viewpoint and from the viewpoint of field extensions. We show an interesting one-to-one correspondence between the so-called Galois groups and field extensions.

    This course serves as an excellent starting point for the Algebraic Number Theory course (Period V), Elliptic Curves (June), and Class Field Theory (August). In terms of practical applications, algebraic number theory is used in, e.g., cryptography and wireless communications.
    Matteo has organised a Zulip chat where you can ask questions outside the lectures and share ideas for exercises and homework. Please join!

    Link to Zulip chat: https://ms-e1111-galois-theory-2023.zulip.aalto.fi/join/35wzgtfbn2jcdzbfpt2gunqf/

    Course staff:

    Responsible teacher: Camilla Hollanti (first week lectures Pavlo Yatsyna)
    TA: Matteo Allaix

    Tentative lecture schedule (by Stewart's chapters):

    Week 1: Ch. 4-5
    Week 2: Ch. 6, 8.5-8.6 (read Ch. 7 and rest of Ch. 8 on your own)
    Week 3: Ch. 9 + Colloquium talk by Ivan Blanco Tue 15:15 Room U6
    Week 4: Ch. 10-13
    Week 5: Ch. 14-15, Ring learning with errors (RLWE) and applications to post-quantum lattice-based crypto
    Week 6: Ch. 18, Hamiltonian quaternions and applications to wireless communications