MATH 426 - Abstract Algebra II Exam II Math Presentation Ideas INSTRUCTOR: Prof aBa GOAL: For Exam II grade, you will be assessed by a Math 426 topic presentation that you will deliver at the Math Retreat Day. You are encouraged to pick a topic from those listed here. If you have a topic not on this list that you would like to discuss then come to my office and talk to me about it. PRESENTATION GROUP SIZE LIMITATIONS: Ideally, you will present with your \Collaborative HW" group. Or you may speak alone or join a different group. At most three people in one presentation group, and EVERYONE in a presentation must speak at their presentation. PRESENTATION LENGTH: Each talk should be around 15 minutes with some time for questions from the audience at the end. You cannot go over 20 minutes which is the full length of the \short" Math Retreat talk intervals. WRITE-UP TO TURN IN BEFORE MATH RETREAT: To assist you all in not waiting super last minute to start working on this Exam 2 Presentation on April 29th, on our Wednesday class on April 24, you will turn in a write-up about your presentation topic. This may include (but is not limited to) writing what you will speak about, new things you learned, and how the chosen topic is directly related to Abstract Algebra. This must be a minumum of 2 full pages typed (preferably TEX'ed). This portion will also contribute to your Exam 2 grade. TALK SLIDES: After the talk, you will send me a PDF of your slides. These will be graded for their quality and preparedness. An example of slides from a short talk I gave is on our webpage. COURTESY TO YOUR OTHER CLASSMATES: You are required to attend your other classmates' talks. This will accomplish two goals: (1) you will learn what they have researched, and (2) it is part of your grade in Exam II. If an extenuating circumstance prevents this, let me know. Very cool ideas directly related to Math 426 1.( Gauss' proof for when a unit group is cyclic) It is well known that the multiplicative group of p q p `q units, denoted U n , of the additive group of integers modulo n, denoted Zn; , is cyclic if and only if n equals 2, 4, pk, or 2pk, for p an odd prime. Discuss the history and interest in this problem and also provide a proof of this classic amazing fundamental result! 2.( Class Number 1 Problem) The class number of a number field is an abelian group which measures how far the ring of integers is from being a principal ideal domain. Which imaginary quadratic fields ? Qp Dq have class number 1? 3.( Euclidean number fields) Some number rings are Euclidean, some are not. What is known? 4.( Euclidean algorithm for quadratic fields) Find all imaginary quadratic fields which are Euclidean. 5.( The exceptional automorphism of S6) Give a description of the exceptional automorphism of S6. 6.( The monster group) What is the monster group? How can it be described, and from where does it arise? 7.( Rubik's cube group) Describe the Rubik's cube group in terms of simpler, smaller groups. There is a lot about this on the web. Also or alternately, you can research other group theory-related aspects of the Rubik's cube. 8.( Finite simple groups) Discuss a class of finite simple groups. For example, prove that the group p q ¡ P SLn Fq are simple whenever q 4. p q p q 9.( The group SL2 Z ) What can you say about the group SL2 Z ? What are generators and relations? 10.( Inseparable extensions) What is an inseparable extension of a field? 11.( Kronecker-Weber theorem) Any abelian extension of Q is contained in a cyclotomic field. Discuss some aspects of the proof. 12.( Frobenius' Theorem) Prove that any finite-dimensional division algebra over R is isomorphic to R, C, or H. 13.( Wedderburn's \Little" theorem) Prove that every finite division ring is commutative (and hence a field). NOTE: There is a beautiful proof of this fundamental result by Ernst Witt in 1931. 2 Highly recommended topics 1.( Galois group of a field extention) Given E and extention of a field F , what is the Galois group of E over F . Give some examples too. Perhaps also give some cool application(s) of Galois theory (do some REAL DIGGING to find some cool application). 2.( Ruler and Compass constructions) Three problems from antiquity (time of ancient Greece in particular) have deep connections to Math 426. They are the problems of • Squaring the circle - draw a square with same area a given circle • Doubling the cube - draw a cube with twice the volume of a given cube • Trisecting the angle - divide an angle into three smaller angles all of the same size using ONLY the basic tools of a ruler (with no measurement markings) and a compass. 3.( Cyclotomic polynomials) The cyclotomic polynomials give the factorization of xn ¡ 1 in terms of polynomials which are irreducible over Q. What are these polynomials, and how dothey bear on the p q th arithmetic of the number field Q ζn obtained by adjoining a primitive n root of unity ζn? 4.( Sum of two squares) Give a nice proof that a prime p can be written as the sum of two squares p x2 y2 if and only if p 2 or p 1 pmod 4q. This presentation might have you exploring a very popular Euclidean domain called the Gaussian integers Zris (depending on what version of a proof you find and use, or come up with yourself). 5.( Insolvability of the quintic) There is a formula to solve quadratic equations, and it is called the quadratic formula. There is a formula too for cubic and quartic equations. But there will NEVER exist a formula for degree five equations. Why is that? This problem is explored via Galois theory, and in some respects it is the SOLE purpose that Galois theory was invented. 6.( Mersenne and Fermat primes) This topic is TOTALLY interesting but might be more a number theory thing than an algebra thing (but of course, those two areas are VERY intertwined due to number n ¡ theory revolving around Zn). In any case, primes of the form 2 1 are called Mersenne primes. We can then prove that n must be a prime for 2n ¡ 1 to be prime. Why is that? Well, if a divides b, then you can show that in the polynomial ring Zrxs the polynomial xa ¡ 1 divides xb ¡ 1 (You prove this!), and as a result 2a ¡ 1 divides 2b ¡ 1. That's why 2n ¡ 1 can be prime only if n is prime. A cool history on Mersenne primes can be found at https://primes.utm.edu/mersenne/. On the other 3 hand, primes of the form 2n 1 are called Fermat primes. We can then prove that n must be a power n b of 2 for 2 1 to be prime. Why is that? Well, if a divides b with a odd, then in the polynomial ring Zrxs the polynomial xa 1 divides xb 1 (You prove this!), and therefore 2n 1 can only be prime if n has no odd factors|meaning it must be a power of 2. 7.( Snake Lemma) You will be allowed to do this if and only if it is not on homework and I don't do it in class, of course. A thorough presentation on this will not just be your group doing the very detailed proof, but you should also do some research to tell the audience why people might care about this result and/or what areas of math it arises and how so. 4.