John Jones - MAT 444

MAT 444

• 5/5 Final exam scores and final grades are available above. The high score on the final was 92 out of 90, and the median score was 55.
• 4/22 Class on Monday will be an overview of Galois theory and a sketch of how fields are used to prove the impossibility of certain geometric constructions, as well as the insolvability of the quintic (and what that means).

  Then Wednesday's class is a question/answer session for the final. So bring questions!

• 4/22 Scores for test 2 are available above. The raw score is out of 94 points. The high score was 100 and the median was 64.
• 4/20 Here are solutions to test 2. Question 1c comes from a section after those listed for the test, so it will be treated as extra credit.
• 4/18 There are more problems posted on the homework page for practice. As of today, we will have finished the content of Section 28 (parts of which were deferred). So, the content for the final are sections: 1-6, 7 (but not Cayley digraphs), 8-10, 12-14, 17, 22-24 (but not solving congruences $ax \equiv b \pmod m$), 26-28, 30, 31, 34, 35 (up to Corollary 35.5).
• 4/13 On Monday, we will start with the statement and proof of Eisenstein's criterion. That will be the last material covered on the final. We will continue with some material on fields which is both a logical next step, and makes use of some of the material recently covered about rings.
• 4/11 The homework page has problems and solutions on a ring which is not Noetherian.

  In class, someone asked for elements in an integral domain which do not have a gcd. The ring considered in class, $\mathbb{Z}[\sqrt{-5}]$, gives an example. The elements $a=6$ and $b=2\cdot (1+\sqrt{-5})$ are such an example. Since 2 and $1+\sqrt{-5}$ divide both, the gcd (if it exists) must be a multiple of both. For starters, the norm of the gcd is then a multiple of both 4 and 6, i.e., a multiple of 12. On the other hand, it is a divisor of $a$ (so its norm divides $N(6)=36$) and is a divisor of $b$ (so its norm is a divisor of $N(b)=24$). So, if $g$ is a gcd, $N(g)$ is a multiple of 12 and is a divisor of $\gcd(24,36)=12$, so $N(g)=12$. One can check that there are no such elements.

• 4/9 The homework due on 4/11 is the last one to be turned in this semester. Problems from later sections will still be posted to the homework page to be practiced on.
• 4/6 For the current homework, you may use that if $F$ is a field, then $F[x]$ is a UFD. We proved the uniqueness in class (for any PID, and that $F[x]$ is a PID since it is a Euclidean domain), and I verbally sketched how to prove existence for this ring (do a strong induction on the degree of the polynomial: if it is irreducible we are done; if not, the two factors have lower degree and we can apply the inductive hypothesis to each factor).
• 4/6 Test 2 is in 2 weeks. It will cover from the end of the test 1 through the homework due on 4/11. That is, it covers Sections 14, 17, 22-24, 26-28, 30 (but only the parts of 28 which we covered in class and/or homework).
• 3/23 Here are solutions to test 1.
• 3/21 Scores for test 1 are available above. The high score was 104 (which counts as 104%), and the median was 71.
• 2/28 The link on the left for the handout on conjugation in $S_n$ is now active.
• 2/28 Dates for tests have been corrected below and on the syllabus.
• 2/26 Test 1 is coming up. It will cover from the beginning of the course through Section 13 (skipping Cayley digraphs and plane isometries). Keep in mind that we assume that you are doing all of the homework from the homework page, not just the ones which are turned in.

  Problems will be drawn from the following types:

  • Defining terms (e.g. group, subgroup, isomorphism) and stating theorems (e.g., State the classification of cyclic groups).
  • Computations, (e.g., computing with permutations, listing finite abelian groups of a given order, counting $n$-torsion elements in a finite abelian group)
  • True/false, like you have seen in the homework, but with justification.
  • Proving theorems from the course (e.g., prove that in a group, inverses of elements are unique, or prove from the definition of subgroup that if $H\leq G$, then $e_G\in H$). These could include homework problems from the course.
  • Proofs of statements you haven't seen before. These are like homework problems, but not the same problems.
  Not all types of problems will necessarily appear. The main emphasis will be on proofs.
• 2/23 The link on the left for groups of order 8 is now active.
• 2/14 The link on the left for groups of order 6 is now active.
• 2/11 The syllabus has been updated to correct the time of the final exam to match the ASU final exam schedule.
• 1/31 Here is some information on writing proofs for this class.
• 1/16 Some students have had problems accessing the online version of the textbook (which is needed to even see the homework problems). So, the first assignment will not be due until Monday, 1/24. In general, Monday will be our homework day.
• 1/13 Here is a page with information on expectations in this course for writing proofs.
• 1/5 The link to the left with an outline of course topics is now active
• 1/3 The links to the left for the syllabus and for binary operations (the latter under pdf handouts) are now active.
• 12/11 The textbook for this course, A first course in abstract algebra by Fraleigh, will be available online for free through canvas. You are welcome to purchase a paper copy as well. If you want a paper copy and cannot find the 8th edition, then the 7th edition is fine. Just be aware that the homework exercises are by number in the 8th edition, but that can be looked up on the free online version of the book.
• About this course

  This is a theorem/proof oriented course which covers both groups and rings. The pace is very fast and the content is for most students, fairly abstract. The courses MAT 440 and MAT 441 each cover roughly half of the material from this course in one semester. They are essentially over-the-counter strength abstract algebra courses whereas this one is prescription strength.

• Exams:
  • Test 1: March 16
  • Test 2: April 20
  • Final: May 4
• Office hours: MW 10:15-11:15 and by appointment