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TATA55 Abstrakt algebra

Matematik och tillämpad matematik

6 hp

Jan Snellman


Senaste nytt/Latest news

  • 2020-02-03: Results are registered. Your exams can be fetched from my shelf in the bureau in the corridor outside my office. Solutions for batch 4 is on the course homepage; batch 5 will appear later this week.
  • 2020-01-24: I have now graded the hand-in exams. I will register the results on Monday next week. All students that have handed in all exercise batches have at least a passing grade, congratulations!

    Since there are still a few students that have been granted an extension and will hand in their exercises next week, solutions of the exercises will not appear on the course homepage yet.

  • 2020-01-17: In exercise 1(ix) and 1(x) of batch 4, this ring was supposed to refer to R, not R/I, which occurs in 1(viii). I inserted 1(vii) and 1(viii) without changing the phrasing.

    Since nearly all that have attempted these exercises have interpreted them to mean that we are looking at monomial ideals in the quotient ring R/I, rather than in R, I have decided to just remove these exercises from consideration.

    The maximum points possible for batch 4 is thus 18.

    Mea culpa.

  • 2019-12-18:
    • If Batch 4, exercise 2(vi), is too hard, do it for the case g(x)=x-r and h(x) = x-s.
    • In Batch 4, exercise 1(iii), an antichain wrt divisibility is a set of monomials such that for any two different monomials from this set, none divide the other.
    • In Batch 4, exercise 1(x), maximal monomial ideal means maximal ideal that is also a monomial ideal.
  • 2019-12-17: The hand-in-date of Batch 4 is January 15, 2020. Batch 5 is due January 20. The calculation of splitting fields in Batch 5 can be tricky, so
    • You may refer to all results in Judson and Svensson, without proof.
    • You may refer to bold claims I make in the lecture notes, without proof.
    • You may check your result using a computer. You should calculate the splitting fields by hand, though, and calculate the degrees of the extensions carefully, for instance by considering appropriate itermediate fields.
    • To check your splitting field degree using SAGE, use something like
      R.< x > = PolynomialRing(QQ)
      f = x^3-1
      This will produce an irreducible polynomial g so that the splitting field is the quotient of Q[x]/(g). Again: this is a tool for you to check your results, but you should calculate the splitting fields by hand.
    • You can also do
      ro =f.roots(QQbar)
      a = ro[0][0]
      b = ro[1][0]
      to find the zeroes.
  • 2019-12-02: We changed the date for the hand-in of batch 3 to December 6. This is this Friday!
  • 2019-12-02: The webserver is back to normal. The lecture notes should be accessible.
  • 2019-11-22: I've graded batch 2 and will hand them out on tuesday next week. If you received less than 11p, there are extra exercises to compensate with.
  • 2019-11-12: On Friday Nov 15 we will have a computer laboration on group actions. This laboration will help you with Batch 3 of the hand-in exam (but Batch 3 can certainly be solved without a computer, only with much more work).

    Bring your laptop if you have one, and look at the instructions for using the computer algebra system SAGE. Basically, install SAGE if you are running Linux, otherwise you will need to use a web browser to compute in the cloud, using CoCalc. This is free of cost, but involves some signing in and administrivia; please check before the laboration that you are able to run SAGE somehow.

  • 2019-11-08: The class on Tuesday Nov 5 decided, unanimously and uncoerced, that Batch 2 of the hand-in exam is to be handed in on Friday Nov 15. You can hand in your assignments in class, place them in my shelf in the piece of furniture in the corridor outside my office. I this is inconvenient, you can also mail them to me, subject TATA55 BATCH2 2019, attach a pdf or gs document (no word).
  • 2019-11-08: The cut-off points for the various grades are: 45 percent for passing grade (3), 60 percent for grade 4, 75 percent for grade 5.
  • 2019-10-17: I have graded batch one of the hand-in exams. A solution sketch is available, and the exams themselves can be fetched from my shelf in the math department corridor in the B-house, in a folder labeled TATA55 out. You can (perhaps) check your results in LADOK, I tried to enter a column for batch one.
  • 2019-01-22: Solutions to the last batch of exercises.
  • 2018-12-13: I have constructed a draft of the third and final batch of exercises. They are divided into two parts, one part consisting of somewhat "standard" problems, and a second part with more elaborate problems. To get a passing grade for this third batch, it is enough to get half of the possible points for the first part. Those of you that have been informed that you should "pass batch 3 with some margin" in order to pass the course, should attempt at least one of the problems in the second part, as well.
  • 2018-12-04: Exercise 3a was correct after all! One apology retracted!
  • 2018-12-03: I have made available the solutions to batch 2. There were quite a few misprints in this batch, for which I apologize. I will try to grade leniently whenever I see that your problems was cause by my mistakes... In exercise 1, "in" should be "of". In exercise 3a, S should be assumed to be closed under taking inverses. In exercise 7b, B=UC should read B=CU. Again, apologies.
  • 2018-10-22: I have constructed the next batch of exercises. This is a preliminary version.
  • 2018-10-18: Some comments on the exercises:
    • In exercise 1, some of you forgot the I asked for positive integer solutions.
    • In exercise 4b, the map f refered to is a general map, not the one from 4a. In 4c, some were confused about "eventually periodic". Please ask me if the formulation of the exercises is unclear! A sequence (x_i) is eventually periodic if there exists N,n such that for all i larger than N, s_(i+n)=s_i.
    • In exercise 5, an element z has order n if z^n=1, and z^k is not 1 for k in (0,n). So the solutions to z^n=1 have orders which are n or which divides n.
    • In exercise 6, I asked about the possible orders of permutations in S5, not how many such there are.
    • In exercise 9, some of you refered to a "theorem" that I apparently stated when lecturing: if x,y commute and have finite order, the the order of xy is the product of the orders of x and of y. This is false in general, and the point of the exercise is to explore this.
  • 2018-10-18: The solutions to Batch 1 of the exercises have been posted. I am still grading away.
  • 2018-09-13: I have been bedridden with influenza, so the first two lectures have been cancelled. The course starts tomorrow. We will have lectures tomorrow and next week, and then we will dedicate some of the scheduled lecture time to sessions where we discuss and work thrugh exercises. In the TIMEDIT link above, some lectures are classified as "Föreläsning" and some as "Lektion"; however, due to the disruption caused by the late start of the course, this information is to be disregarded. As stated above, the course will start with two lectures, then we will see.
  • 2017-01:27: I have started grading Batch 3 of the exercises, and hopefully I'll be done by the end of next week. I apologize for the delay. A solution sketch is now available. It includes the weighting of the individual exercises.

    Some remarks:

    • The algebra generated by the 4x4-matrix C (inside the non-commutative algebra of 4x4-matrices) is commutative, since various powers of C commute. It is however not necessarily a domain, since there are zero-divisors among square matrices. In our case, (C-4I)(C-4I)=0, so our algebra has zero-divisors. It is absolutely not a field!
    • A polynomial of degree 3 or less is irreducible iff it has not linear factors, hence iff it has no zeroes. However, a polynomial of degree 5 can factor as a product of something of degree 2 times something of degree 3, with neither of the factors having zeroes!
    • The formulation of the question regarding monomial ideals was evidently too terse. An ideal I in R is generated by monomials if there is some (not necessarily finite) set S of monomials the generateS, that is , for any f in I there is a finite subset S_0 of S such that f is a R-linear combination of elements in S_0.

      As a matter of fact, any ideal in C[x,y] which is generated by some set S is generated by some finite subset of S (this is Hilbert's basis theorem), but this is not something which is necessary to solve the exercise, nor is it something that you can assume without proof!

      The bijection (a,b) -> x^ay^b gives a bijection between N^2 and the set of monomials in C[x,y]. It induces a bijection between monoid ideals in N^2 and monomial ideals in C[x,y] by mapping a set of exponents in N^2 to the monomial ideal generated by the corresponding set of monomials in C[x,y]. Different things.

      The (admittedly tedious) check that the above bijection maps sums to sums need to involve the following step: check that the monomials occuring in the support of elements in the ideal I+J in C[x,y] is precisely those monomials that occurs in the support of elements in I or in the support of elements in J.

  • 2016-12-14: Some remarks regarding Batch 2:
    • Exercise 2e, misprint: just show that it is an automorphism
    • Exercise 3b, misprint: automorphism should be homomorphism
    • Exercise 4, misprint/omission: Naturally, I meant the number of non-equivalent colorings.
    • Exercise 2f, one should list the automorphisms, and also show how the multiplication (composition) works, to see that the first Aut is isomorphic to Z_n^x.
    • Exercise 3d, note that while N^ is normal in NxK (with the strange new product), K^ is not necessarily normal. Furthermore, with the new strange product, elements in K^ does not necessarily commute with elements in N^.
    • Exercise 5, For this exercise only, I have awa rded full points for correct answers without motivation. To give a complete classification of the conjugacy classes in A(n) for a general n, see e.g. Scott, Group Theory, Thm 11.1.5.
    • Exercise 6: denote the conjugacy class of g by Cl(g). This is not a subgroup; to form the subgroup N that it generates we need to take all finite products of conjugates x_igx_i^{-1} of g and conjugates x_jg^{-1}x_j^{-1}.
  • 2016-12-14: We decided jointly that Batch 3 of the hand-in exam is due January 15.
  • 2016-10-04: I have been bedridden for two weeks, but now I am back on my feet! Thus, the lecture on wednesday 5 Oct will be held!
  • 2015-02-05: The hand-in exams have been graded. Some comments:
    • A polynomial (over Q) of degree 3 has a splitting field of dimension at most 3!=6 as a Q-v.s.
    • In exercise 3, one should really form the 6x7-matrix of coefficients of powers of delta if one wants to use the nullspace to calculate the minimal polynomial, since it has degree 6.
    • In exercise 4, I meant F^*, but those of you that answered the question as it was written has not been penalized for your lack of mind-reading skills.
    • In exercise 6, check the corresponding exercise in Svensson, page 476. In particular, one should find the degree of the algebraic, real number which is the x-coordinate of a vertex of the heptagon. This is algebraic of degree 3 (not 6).
  • 2105-01-30: I have been ill, and have not been able to grade the final batch of hand-in exercises. They will be graded next week.
  • 2014-10-23: Missprint in ex2B2: should read (g1,h1)*(g2,h2)=(g1*g2,h1*h2)
  • 2014-10-09: Some comments on HIEB1:
    1. If you embed a group G into a larger group B, then a necessary but not sufficient condition for two elements in G to be conjugate is that their images in B are.
    2. The crux of question 3b is to show that every linear isometry that maps the regular n-gon into itself must map each vertex to (some other ) vertex. The easiest argument is to note that the vertices are extremal w.r.t. distance to the origin.
  • 2014-10-09: In the last lecture, I was confused about the subgroups of size four of the dihedral group of a hexagon. This short article shows that the groups in question are {1,r^3,s,sr^3},{1,r^3,sr,s^4},{1,r^3,sr^2,sr^5}.
  • 2014-09-25: Question 3d in Batch 1 is incorrect; show instead that when n (the number of vertices) is odd then all reflections are conjugate.
  • 2013-12-16: Inlämningsuppgift 3 inlämning 20 dec; omgång 4 inlämning 15 jan? Diskuteras på sista lektionen.
  • 2013-12-04: Lösningsförslag till uppgift 7 på omg 2: Sylows första ger oss en delgrupp H med p^(r-1) element. Låt G verka på X, mängden av vänster sidoklasser till H genom g. (xH) = (gx)H. Vi har att |X|=p, och vi får en representation F: G -> S_X = S_p (ungefär som i Cauchys sats, men nu är det sidoklasser som permuteras, inte element). Naturligtvis är N=ker F en normal delgrupp till G. Sätt s=[G:N]. Då är även s=|Im F|. Vi har följaktligen att s delar |G|, samt att s delar S_X, dvs s|p^r, s|p!. Detta ger att s=p. Alltså har N index p i G.
  • 2012-11-14: Inlämningsuppgift omgång 2, uppgift 2: Judson visar att om S_2 verkar på växlingsfunktioner av två variabler, erhålls 12 banor. Behövs inte visas! Hur många banor får man, med 3 variabler, delgruppen generarad av 3-cykler? Hela S_3? Ge antalet, lista ej banorna. Resten av uppgiften verkar korrekt.
  • 2012-10-09: Inlämningsuppgifter omgång 1, uppgift 2. T_1 skall bestå av tvåcykler samt identitetspermutationen.
  • 2013-01-14: Inlämningsuppgifter omgång 3, uppgift 4: Tryckfel: Z_x skall vara Z_2. Fel i uppgiften: alpha ej generator för multiplikativa gruppen (den har ordning 9). Visa istället att alpha+1 är en generator.
  • 2013-01-14: Inlämningsuppgifter omgång 3, uppgift 6: irreducibelt är fel, skall stå moniskt.

Sidansvarig: Jan Snellman
Senast uppdaterad: 2020-04-10