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# TATA55 Abstrakt algebra

Ämnesområde

Poäng
6 hp

Examinator
Jan Snellman

Schema
Timeedit

## Senaste nytt/Latest news

• 2023-12-02: Started grading, found typo in Solutions to hand-in exercises batch 4. Now corrected, 2c should have
• 2023-12-01: B4 might not be swiftly graded, busy upcoming week.
• 2023-12-01: Lecture 4 dec moved to 19 dec. B5 due dec 21.
• 2023-12-01: There was a misprint in B5, last exercise, part a. Now fixed.
• 2023-11-15: Preliminary version of hand-in exercises batch 5 now available for perusal. Will be due some time in december. there will be a batch 6, due in january 2024.
• 2023-11-08: Grading of batch 3 done, grap your corrected exams from my shelf. Due date of B4: last day of november.
• 2023-11-06: Solutions to hand-in exercises batch 3 now available. Grading is in progress.
• 2023-10-18:
1. The due date for hand-in exercises batch 3 is November 3.
2. Yesterdays lecture, I was confused about colorings of the square. D4 contains the identity, 2 rotations with cycle structure {1,3}, contributing 2k^2 colorings with k colors, and 1 rotation with cyle structure {2,2}, contributing k^2 colorings with k colors. So far, so good. D4 has 2 reflections through opposite vertices, with cycle structure {1,1,2}, contributing 2k^3 colorings with k colors, but (I missed this distinction) also 2 reflections through midpoints of opposite edges, with cycle structure {2,2}, contributing 2k^2 colorings. Thus there are 1/8(k^4+2k^2 +k^2 + 2k^3+ 2k^2) colorings, which evaluated at k=2 becomes 6 non-equivalent colorings.
3. If you want to know about applications of group theory to counting problems, you can read the manuscripts refered to on the course homepage, e.g. this paper by Huisinga.
• 2023-10-12: Preliminary versions of hand-in exercises batch 3 and hand-in exercises batch 4 are now available.
• 2023-10-06: I have constructed solutions to HT2023 b2 but have not yet had the time to start grading them.
• 2023-10-04: Since I have a lot of grading in another course, grading of batch 2, and writing solutions for batch 2, will be somewhat delayed. I will have it done before next lecture, though. Batch 3 will hopefully also be constructed before then.

If, in the future, you e-mail me your hand-in exercises, please give your message a title containing TATA55. If you need to receive feedback/results electronically, please state that; by default, I print your submissions and hand that out during the next lecture, then store them on the shelf in the cupboard in the corridor outside my office.

There is a collection of exercises on the course homepage that I encourage you to work on.

• 2023-09-19: due date for batch 2 of hand-in exercises changed to Oct 3.
• 2023-09-12: I slightly modified Hand in exam batch 2.
• 2023-09-11: Hand in exam batch 2 is due september 26.
• 2023-09-04: Hand in exam batch 1 is due september 19.
• 2023-01-19: Solutions to Hand in exam batch 4 now available. I am almost done grading.
• 2023-01-07: I have finished writing down solutions to Hand in exam batch 4, and want to give some hints.
• Part 1, Exercise 1: Look up the Cayley-Hamilton theorem and "minimal polynomial of a matrix" if you have not encountered those before
• Part 2, Exercise 3: This is an open-ended, very difficult exercise. Do what you can. For which primes p is f(x) irreducible over Zp? If it is irreducible over Zp, for which k is it irreducible over GF(p^k)? You may certainly use your computer to form some conjecture, some implications may not be impossible to prove.
• Part 2, exercise 4: If you find some algebraic relation over Q that a satisfies, then to find the degree you want to show that this relation is of smallest degree, i.e. irreducible.
• Part 2, Exercise 6: An integer m is a perfect square if there is some other integer s such that s*s=m
• Part 2, Exercise 7,8: Finding the splitting field means finding the degree over the prime subfield, and giving the splitting field as tower of simple algebraic extensions. You should give some argument to show that no smaller extension will work. You may not use theorems on discriminants without proving them, first. The simple-minded method in the computer lab may be used: make a Kronecker extension and see if the polynomial splits there, if not, make another Kronecker extension.
• 2022-12-20: Rejoice: everyone passed B3, only B4 left! Recall that the due date for B4 is jan 17 2023. Please report missprints/errors/unclear formulations in B4 to me.
• 2022-12-20: Solutions to Hand in exam batch 3 now available
• 2022-12-16: There are a few misprints in Hand in exam batch 3. The due date is wrong; was originally dec 15 but we changed it to dec 16. Exercise 1 of part 1 mentions ordered pairs, should be unordered pairs, thus E2 = {{ (f(u), f(v)} : {u,v} in E1}
• 2022-12-09: A draft of Hand in exam batch 4 is now available. Please check it for errors, omissions, dubious double meanings, reasonableness... Also ponder the following question: do I have to provide some SAGE code for part one, or can you write that youurselves?
• 2022-11-23: A draft of Hand in exam batch 3 is now available. Please check it for errors, omissions, dubious double meanings, reasonableness...
• 2022-11-23: We decided in class that Batch 3 of hand-in exams are due December 15, 2022, and that Batch 4 is due January 13, 2023. Batch 3 is not yet constructed, but will feature some computer experiments from LAB2
• 2022-11-19: Solutions to Batch 2 are now available. I have just started grading, will probably not be done by monday.
• 2022-11-07: We decided that Batch 2 is due Thursday nov 17. If you do not want to hand in your work on the lecture on November 15, you can either e-mail me or put it on my shelf
• 2022-11-02: We decided on Nov 10 as the due date for hand-in-exam B2, but since yesterdays lecture was cancelled, maybe you need more time? Let's discuss this next time we meet (nov 7). It will of course be OK to hand in B2 early.
• 2022-11-01: Todays lecture is cancelled, I am ill.
• 2022-09-28: You can now consult the solutions to hand in exam B1. I have graded your submissionns; everyone passed! The following students handed in B1, if you are not on the list but handed something in, contact me. Last name 2 letters/First name 2 letters

BeJo LeBeOl BlSi BrKr ClMa DaFi (LADOK mystery) DePa ErKa GrOs GrDa JoEr KaEl (not registered) PoAc RyCh SkFr

• 2022-09-26: The hand in exams batch 1 has a an error, in exercise 5: it should say "full" binary trees, rather than "complete" binary trees. I mean that every node has zero or two children. In part (d), the product of two "pruned trees" is equivalent to a unique pruned tree, that is what I am after. 5b and 5e are related: what I mean in part b is really "show that there is a finest eq rel that induces associativity". I mean, you could use the eq rel with a single part, and get the trivial monoid; that is not what I meant. Furthermore, in 5e it should say "tilde is a refinement of" rather than the other way around.
• 2022-09-16: On sept 26 there will be a computer algebra lab, in the classroom (R36). Bring your laptop! You can certainly work in groups of two, so as long as half of you bring something with a keyboard, and either sagemath installed or wireless ethernet capable (eduroam) then we are golden.
• 2022-09-16: Take-home exam batch 1 due sept 27, see relevant course homepage
• 2022-01-12: Grading went swifter than expected, your results are now registered.
• 2022-01-11: Solutions to batch 4 has been posted.
• 2021-12-30: I have answered some questions regarding batch 4, and given the following hints:
• To find the minimal polynomial of an element in a finite extension of degree n, form the first n+1 powers of said element, they are linearly independent (see Judson thm 21.15).
• The process of converting a (rational) generating function of a sequence to an explicit formula, via partial fraction decomposition, is described in the lecture notes (last lecture). This works very well over finite fields.
• 2021-12-14: I made corrections to batch 4 after errors were pointed out. Recall that this batch is due januari 10, 2022. You may e-mail me your submission, or place it in my personal shelf in the dedicated bureau in the corridor outside my office. I will answer e-mail queries on batch 4, both regarding the interpretation of the exercises, and about SAGEmath. A tip: do some exercises in the textbook, in particular about splitting fields, before you start on the hand-in exam!
• 2021-12-14: Latest version of lecture 15 includes treatment of difference equations over finite fields.
• 2021-12-10: solutions to batch 3 .
• 2021-11-30: Some minor errors in HT2021 Omgång 3/Batch 3 corrected.
• 2021-11-30: We, those assembled in BL34 on this november afternoon, decided on the early date of January 10, 2022 as the hand-in date for batch 4 of the exercises (the last batch). Let no one who was not there object!
• 2021-11-30: Some notes from todays problem-solving session is available on the lectures page.
• 2021-11-22: You can now look at the solutions to batch 2 .
• 2021-11-16: We decided that Batch 3 of the hand-in exercises should be handed in on December 7. The lecture on November 30 will be replaced by a problem-solving seminar. We'll do some exercises in detail. You can also ask for clarifications (and hints) on Batch 3.
• 2021-10-15: Correction HT2021 Omgång 2/Batch 2. Exercise 2: stabilizer subgroups just for i.
• 2021-10-14: Correction to batch 2: ex 2: matrices should be complex
• 2021-10-07: I have constructed solutions to HT2021 Omgång 1/Batch 1.
• 2021-09-24: Batch 1 due date changed to october 5.
• 2021-09-13: Lectures/lessons in october will be in classrooms. Batch 1 due sept 30.
• 2021-09-01: I have concocted a preliminary version of hand-in-exam HT2021 Omgång 1/Batch 1. Please check for readability/reasonability/making of sense.
• 2021-08-27: I tried e-mailing the TEAM to see if everything worked, and also to see if there are any foreign students, to determine whether the course will be given in swedish or english. It seems that the TEAM page, rather than the team members, received the letter? (Never mind, copying and pasting all 13 names into to the receiver field worked).
• 2021-08-25: The course WILL be given online, via the TEAMS group TATA55_HT2021. I have tried to find the registered students via LADOK and added them to the TEAM; e-mail me if you have not been added this week.
• 2021-08-23: The course during autumn term 2021 will probably be given virtually via TEAMS. I will decide presently and inform students here.
• 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
f.splitting_field(x)
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!
• 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