Ämnesdisposition

  • Allmänt

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    • Instructor: Sofia Tirabassi
      • Email: tirabassi@math.su.se
    • TA: Benedetta Andina
      • Email: benedetta.andina@math.su.se
      • The lecture dates and times are as reported on TimeEdit.
    • Textbook: Abstract Algebra by Dummit and Foote (mostly Chapters 13-14, see sections posted below).
    • Prerequisites: Mathematics III: Abstract Algebra. This course also uses the book by Dummit & Foote, and reviewing material from Chapters 1-4, 7-8 could be helpful for Galois theory. We will occasionally review and fill in details of important things from those chapters directly in class and exercise session. 

  • Examination & Grading

    1. Final Written Exam: 

    • Out of 30 points 
    • You need 12.5 to pass. 
    • 26.5-30 is an A, 23-26.5 is a B, 19.5-23 is a C, 16-19.5 is a D, 12.5 to 16 is an E. 
    • (There is no oral exam.)
    • It is mandatory to register for the exam in advance on Ladok. Please remember to do so on time!

    2. Homework bonus: 

    • There will be 3-5 homework assignments posted on this course website. 
    • You also submit the homework on the course website. 
    • We average the score out of 3 bonus points to be added to the final exam. 

    Example 1: You get 24.5/30 on the final exam and don't do any homework. Your final grade will be B.

    Example 2: You get 22/30 on the written exam, there were 3 homework assignments and you got 40/60, 35/60 and 60/60  on them. Then your average homework score is 45/60, which makes 2.25/3. So your grade with HW bonus is 22+2.25=24.25 which is again a B.


  • Some Recent Exams

  • Other texts (not required for the course)

    Here are some other texts, either directly about Galois theory, or about its relationship with other subjects, such as algebraic topology, number theory and/or algebraic geometry. These texts are mostly for those of you who are curious to explore beyond the course itself. These texts are not required and should not scare anyone.

    • URL icon
      Undergraduate Algebra by S. Lang URL

      Slightly more elementary text which covers less material than Dummit & Foote, but covers a substantial part of the course, often using slightly different arguments. For example, I like that Lang puts a stronger emphasis on embeddings than Dummit & Foote, see section VII.2. Also VIII.6 "Where does it all go?" is a nice exposition/motivation for how Galois theory leads to current research in the Langlands Program. (If the link does not work directly, then go to https://link-springer-com.ezp.sub.su.se/, use your SU login and then search for Lang Undergraduate Algebra and you will be able to download the book for free via SU access)

    • URL icon
      Exploratory Galois theory by J. Swallow URL

      Another introduction to Galois theory, seems roughly at the same level as Dummit & Foote, maybe some more examples to consider.

    • URL icon
      Algebra and Galois Theories by R. Douady and A. Douady URL

      Tries to present simultaneously the Galois theory of field extensions and the analogous theory of covering spaces in topology.

      You might need to go to https://link-springer-com.ezp.sub.su.se/ first, login, and then search for "Douady Galois theories"



    • Fil icon
      Idees Galoisiennes by Y. Andre Fil

      Expository article in French about the history of Galois theory and its role in the development of modern mathematics. If there is sufficient demand I will consider translating it.

    • URL icon
      Galois groups and fundamental groups by T. Szamuely URL

      A more advanced text about the analogy and common thread between Galois groups and fundamental groups; this one involves more algebraic geometry and less topology.

      You might need to go to https://mathscinet-ams-org.ezp.sub.su.se/ first, use your SU login, then search for Author = Szamuely and Title = Galois fundamental

    • URL icon
      Algebraic theory of numbers by P. Samuel URL

      Does not seem to be available online via SU login (I'm told you can find it online if you look a bit harder)

      Chapter 6 gives a concise introduction to how Galois theory enters in the arithmetic theory of algebraic numbers, via the decomposition and inertia subgroups of the Galois group, and how this is related to the splitting of primes in extensions.





  • Introduction & Overview

  • Week 1 - Introduction and review

    In the first lecture we overview the course goals and we review some notion from abstract algebra

    • Definition of field extension, degree of the field extension, field extension as vector space. The Galois group of a field extension.
    • Rings of fractions, particularly the fraction field of an integral domain. See Section 7.5 of Dummit and Foote.
    • Unique factorization domains (UFD) See Section 8.3 of Dummit and Foote.
    • Gauss Lemma and Eisenstein criterion on factorization/irreducibility of polynomials. See Sections 9.3.-9.4 of Dummit and Foote.

    Suggested Exercises:

    DF Sec 9.4: 1, 2, 12

    See exercise sheet



  • Week 2 - Field Extension, finite, finitely generated and algebraic

    We start put down the basis needed to understand and prove the fundamental theorem of Galois theory . The main goal of today is to give a charracterization of finite extension. to do that we need to cover the following topics

    • Algebraic and trascendental extension
    • Simple extension and finitely generated extension
    • minimal polynomial of an algebraic element, degree of the corresponding simple field extension
    • degrees of towers of field extenstion
    • finite extension is equivalent to algebraic and finitely generated

    References for Topic 1: (a) Dummit and Foote: 13.1-13.2 

    Suggested exercises

    DF Section 13.1: 1, 2, 3

    DF Section 13.2: 1, 3,7, 10, 13


  • Week 3 - splitting field

    Splitting field of a polynomial

    Dummit and Foote Section 13.4

    Exercises: 1, 2, 3, 4.




  • Week 4 -Separability

    Separability

    Dummit and Foote Section 13.5

    Exercises: 1, 2, 5, 8

  • Week 5 - Embeddings

    Combines some material from 

    • Dummit & Foote 13.4 & 14.2 (but only a small part of 14.2)
    • Lang VII.2
    • Lecture notes from Oslo

    Exercises 1, 2, 6 (not the part of Zorn Lemma, and 9 from the notes from Oslo.


  • Week 6 - Finite fields

    DF section 14.3

    Exercises 1-5


  • Week 7 - Root of Unity and cyclotomic extensions

    DF Section 13.6


    Exercises 1-6.

  • Week 8 - The primitive element theorem

    DF 14.4

    Exercises 1 ,2 4, 5

  • Week 9 - The fundamental Correspondence

    DF 14.2

     Exercises: 3, 4, 5, 6, 7, 10, 12


  • Week 10: Cyclotomic and abelian extensions

    Df 14.5

    Exercises 1, 2, 3, 5

  • Week 11: Construction by straightedge and compass

    Df 13.3 and 14.5 *fir the construction of ghe regular n-gon.


    For exercises look at Extra Practice

  • Week 12: Solvability by radicals

    DF 14.7

    Exercise 2,3,4


  • Week 13: Computing the Galois group I

    DF 14.6 and 14.8

    Df: 14.6:  2, 4-10, 13, 14,16
    Try to use all the tchniques you know to exclude variuos possible Galoi group, not just discriminant and resolvenat  algotrithm


  • Week 14: Computing Galois groups II

    DF 14.6, 14.8


    Exercises DF 14.8: 1 2, 3, 4, 5, 6, 7, 8, 9, 10

  • Week 14 - A proof of quadratic reciprocity

  • Artin's Example in Lang's Algebraic Number Theory

    [Optional: Not required for the course] See p. 121 of Lang's "Algebraic Number Theory". The book is available with an SU login. You need to make it to here: https://link-springer-com.ezp.sub.su.se/book/10.1007%2F978-1-4684-0296-4