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UPDATED (July 5): MT1 will cover the topics from the first 83 pages of Herstein, as well as the material we have covered in class and on the homework. You don't need to know about the complex numbers (section 1.7) or the Euler phi-function and Euler's Theorem (pg 62,63) or Cayley's Theorem (pg 69). You don't need to know Cauchy's theorem (pg.80). The exam won't cover the homomorphism theorems (except for what we talked about (July 5) in lecture about "making isomorphisms from homomorphisms). We'll cover all this material a little later. Review Material: Here are the midterm and the partial solution from a class like this one taught by a colleague of mine. Review Material: Here is another midterm and solutions from another Math 113 class. Review Material Here is last summer's MT1 from the course I taught. Midterm 1 and Solutions Midterm 1 and Solutions |
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MT2 will cover the topics from all of Herstein up to section 4.6 (except 1.7 and anything from Chapter 3 that we didn't talk about in class). It also covers some material that is not in Herstein, namely the material on Symmetries from class, and the material on group actions. It also covers the material from the first 6 HW assignments.
Review Material: Here are the midterm and the partial solution from a class like this one taught by a colleague of mine.You should not worry about question 4. Review Material: Here is another midterm and solutions from another Math 113 class. This exam is a little heavy on the group theory. Review Material: And one more midterm and solutions from another Math 113 class. Review Material: Here are the review questions and solutions for the second exam from the last time I GSI'd this class. You can ignore the blurb at the beginning. Review Material: Here is last summer's MT2 from the course I taught. Midterm 2 and Solutions Midterm 2 and Solutions |
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The Final Exam will cover all the sections from Herstein that we talked about in class. In particular, we covered all of Chapters 1 and 2. We covered all but section 3 in Chapter 3. We covered all of chapter 4. We covered all but section 5 in chapter 5. We covered sections 6.2,6.3,6.4. We also covered my notes on group actions, and some material on symmetries. You should prepare for the final by: writing out and remembering the definitions of the terms we learned this semester, writing out and remembering the statements of the important theorems (try to translate each statement into a short phrase, but also remember the exact state,ent.) Look over the solutions from the homework assignments, especially for those problems that you missed the first time around. Look over the past midterms, and check the solutions, especially for the problems that you missed. Look over the proofs of the important theorems. Look through the review material, and problems throughout Herstein's great book. Review Material: Here are the practice final and the partial solution from a class like this one taught by a colleague of mine. Review Material: Here is last summer's Final from the course I taught. Also check out the exams and solutions I posted as review material from the previous midterms. |
| June 25 | We covered some basics about sets and functions (mappings). Especially make sure that you know the definitions of 'injective (one-to-one)', 'surjective (onto)', and 'bijective' funtions. We also talked about some general strategies of proof: direct proof; proof of the contrapositive; proving two sets are equal (by showing each one is contained in the other); proving a function is invertible (by writing down its inverse), proving an equality (by separately looking at the LHS and RHS). | ||||||||||||||||||||||
| June 26 | We finished the argument that a function is bijective iff it is invertible. We talked about A(S) - in particular we talked about the four axioms (closure, associativity, identity, inverses) and about a useful notation (cyclic notation). We remembered some things about the integers. First, with the operation + we have the same four axioms as for A(S) above (closure, associativity, identity, inverses). Also - divisibility, primes, primefactorization, gcd, Euclidean 'Algorithm'. The two most important things we've done so far are (1) start to admire (and learn and memorize) the four axioms, (2) learn that Euclid's Algorithm gives us a way to write the gcd of two integers, m and n, as a linear combination of m and n i.e. gcd(m,n) = am+bn for some integers a and b. | ||||||||||||||||||||||
| June 27 | We talked about modular arithmetic. Congruence classes/ equivalence classes of integers mod n. Well-definedness of the addition mod n. How to find the inverse of [k]_n if k and n are relatively prime. We also talked about induction. | ||||||||||||||||||||||
| June 28 | We finally defined "Group" - by giving a list of axioms. The axioms were the same ones as we saw for the examples A(S), Z, Z/nZ earliers this week. So all of those are good examples of groups (what is the operation in each case?). We also gave some more examples: (R,+), (Q,+), (C,+), (GLn(R), matrix mult.), (Q^*,*), (R^*,*), (C^*,*), (Un,*), etc. We talked about subgroups, and worked through some examples and non-examples. | ||||||||||||||||||||||
| July 2 | We did a long example about S_4, and then we talked about equivalence relations. In particular, we talked about the most important equivalence relation for this class: for a,b in a group G with subgroup H, a equiv to b iff aH = bH. Cosets, and Lagranges theorem. | ||||||||||||||||||||||
| July 3 | We discussed cosets and some tricky points from HW2 more, and then talked about homomorphisms and isomorphisms | ||||||||||||||||||||||
| July 4 | Holiday! | ||||||||||||||||||||||
| July 5 | We reviewed the ideas of homomorphism and isomorphism, and talked about what it means for two groups to be isomorphic, namely that ALL their structural properties are the same (e.g. either both abelian or both not, either both cyclic or both not, either they both have an element of order three or they both don't, either they both have 17 elements or they both don't, either they both have a subgroup of order 5 or they both don't, etc.) If someone gives you two groups, and asks if they are isomorphimic, you should look at some of their structural properties and see if they are the same. If they are, then look for an isomorphism. If not, then you know right away they are not isomorphic. We showed that all cyclic groups are isomorphic to (Z,+) or to (Z/nZ, +n). We showed that if you start with any homomorphism f:G --> G', you can makeit into an onto homomorphism by restricting the target space so you have f:G --> f(G). Then you can get an isomorphism by defining a new equivalence relation on G that says that two elements of G are equivalent if they get mapped by f to the same point in f(G). We showed that this is the same equivalence relation as from two days ago, and used that information to define a new group homomorphism f: G/Ker(f) --> f(G), which is now an isomorphism. | ||||||||||||||||||||||
| July 9 | Review Session for MT1. | ||||||||||||||||||||||
| July 10 | Midterm 1: 12:10-1:30 in our usual classroom. You will need only your pencil and your thinking-cap. | ||||||||||||||||||||||
| July 11 | I handed back MT1. Here is some information about the way the class did. We finished our discussion of factor groups and the Homomorphism Theorems. I have found some notes I wrote a long time ago on some of this material, and you may find them useful: notes on the First Homomorphism Theorem. and notes about factor groups and the Third Homomorphism Theorem (in the notes, the theorem is called the `isomorphism theorem'). We also started to talk about symmetries. | ||||||||||||||||||||||
| July 12 | Some more on symmetries. Group actions: Here are some notes for today's material (and one thing from Monday) on group actions that's not in Herstein. | ||||||||||||||||||||||
| July 16 | We spent the class talking about a few applications of group actions. Burnside's Formula for counting the number of orbits of an action. We talked about the colorings of the tetrahedron. Here is a solution to an assignment question from the last time I taught this class: it may help you think about group actions, symmetries of the cube, Burnside's Formula; and it is a more complicated version of the same problem (about the necklace) you have on HW4. We used group actions to prove Cayley's Theorem. Cayley's Theorem is in your textbook, and the proof I gave in class is essentially the same as the proof from your book, but expressed in the language of group actions. | ||||||||||||||||||||||
| July 17 | We proved Cauchy's Theorem. The proof from class is the same as the proof from the textbook, except in the language of group actions. We began the material on conjugacy in section 2.11. In particular, we defined the class equation, and we showed that groups of order p^n have non-trivial center. We discussed some ideas for HW4. | ||||||||||||||||||||||
| July 18 | We warmed up with a proof that groups of order p^2 are abelian. I went in detail through the "actions-language" proof of Sylow 1, and then stated Sylow 2 and 3. We did a few examples of how these theorems are useful to find out about groups whose orders have factorizations that are not too complicated. We recalled the definition the direct product of groups, and stated the Fundamental Theorem of Finite Abelian Groups. | ||||||||||||||||||||||
| July 19 | We went through the details of a typical sort of calculation you might want to do using the Sylow Theorems and the Fundamental Theorem of Finite Abelian Groups. This finished up our groups theory for the time being. We covered the material in sections 4.1 and 4.2. This includes the definitions and examples of rings and subrings. | ||||||||||||||||||||||
| July 24 | We talked about homomorphisms, and how they are maps that `preserve' algebraic structure. So ring homomorphisms are maps that preserve all the ring operations - in particular, since rings are abelian groups with some extra properties (multiplication), a ring homomorphism is a group homomorphism with some extra properties. We gave lots of examples and talked about how to find all the homomorphisms from certain types of rings to others. We talked about wanting to make quotient rings, and ended up talking about ideals. We talked briefly about some ideas from HW5 | ||||||||||||||||||||||
| July 25 | We talked about the Correspondence Theorem, and the Homomorphism Theorems. The important theme for this lecture was that the statements of these theorems are easy to remember if you are comfortable with the theorems from our group theory unit, since they are the same - the only difference is that we write them with additive notation (since the rings in question have + as their group operation), and wherever we had `normal subgroup', we write `ideal' instead. | ||||||||||||||||||||||
| July 26 | We talked about maximal ideals, and showed that an ideal I in a commutative domain with unit is maximal if and only if R/I is a field. We restricted our attention to commutative rings, R. We talked about polynomial rings R[x]. We showed that if R has a unit then so does R[x], and that if R is a domain, then so is R[x]. Moreover, we showed that if R is a field, F, then F[x] is really nice! F[x] has a Euclidean algorithm, which gives us gcd's and allows us to show that every ideal is principal (i.e. every I is of the form I = {af | a in R} for some f in R). | ||||||||||||||||||||||
| July 27 | We defined `irreducible', drawing parallel to the idea of prime integers, and stated the theorem that principal ideal domains are unique factorization domains. We showed that in a PID an ideal is maximal if and only if it's generated by an irreducible element. Then we talked about a bunch of tools to help decide whether or not a polynomial is irreducible. (Not in order: Eisenstein, Gauss Lemma, the `highschool' test (i.e. guess and check possible integer factors), the root-finding test (i.e. for polynomials of degree < 4, they're irreducible iff they have no roots), and the reduction mod p test (i.e. if you reduce the coefficients of your integer polynomial mod n, and get a new polynomial of the same degree that's irreducible over Z/nZ, then the one you started with is irreducible.) | ||||||||||||||||||||||
| July 30 | Review Session | ||||||||||||||||||||||
| July 31 | Midterm 2 | ||||||||||||||||||||||
| Aug. 1 | Fields. We talked about the construction of Frac(D) for any domain D. The construction parallels the construction of the rational numbers from the integers. We talked about the sense in which Frac(D) is the smallest field containing D, and you will do more on this in HW7. For a field F, I defined F(a) as Frac(F[a]), and pointed out that it's the smallest FIELD that contains all the elements of F as well as a. Then we Looked at the complex numbers, and in particular constructed them as R[i], which is isomorphic to R[x]/(x^2+1). We talked about some properties of C, namely norms, conjugates, polar form expression and some calculations like multiplication, addition and taking inverses in both polar and cartesian coordinates. | ||||||||||||||||||||||
| Aug. 2 | I handed back MT2. Here is some information about the way the class did. Continued our discussion of fields. Defined characteristic of a field and gave lots of examples of fields that we've seen before. We talked about extensions of fields, and their degrees (namely [E:F] = the dimension of E as an F-vector-space. We showed that if p is an irreducible polynomial of degree n over a field F, then E = F[x]/| Aug. 6 | We defined algebraic and finite extensions, and showed that every finite extension is algebraic. We also showed that if F is an extension of E then the set A={a in F | a is algebraic over E} is a subfield of E and an extension of F}. We showed also that for a tower of finite extensions E subfied of K subfield of L the degrees multiply i.e [L:E] = [L:K][K:E]. And I stated at least that it's not true that every algebraic extension is finite. Aug. 7
| I gave the example Q subfield A subfield C (see above for def'n of A) to show that A over Q is an algebraic but NOT finite extension. As a corollary, we saw that R over Q is not a finite extension (i.e. as a Q vector-space, R is infinite dimensional). We recalled the definition of roots of polynomials, and for any irreducible p[olynomial f in F[x] talked about the abstract construction of an extension field E of F which contains a root of f. Then we stated and proved the theorem that for any degree n polynomial f in F[x] there is a field extension E of F in which f can be factored in to linear factors, and that this extension can be taken to have degree no greater than n!. We called such a field extension a 'splitting field' for f, and mentioned that a minimal such extension exists, and is essentially unique (although we didn't prove these last two claims). We also talked about the miracle of the complex numbers that they are algebraically closed.
| Aug. 8 | All About finite fields. We showed that a finite field K must have prime characteristic, and that it contains F_p (i.e. Z/pZ) as a subfield. We argued that since this makes K a finite dimensional vector space over F_p, it must have q=p^n elements. Then we proved a big theorem about fintite fields. Namely, we showed that "There is exactly one field, K, (upto isomorphism) with q=p^n elements for any prime p and any n>0.". Along the way, we showed that the elements of K are exactly the roots of the polynomial x^q-x in F_p[x]. We also showed that the multiplicative group of non-zero elements of K is cyclic. Aug. 9
| Finite Fields (conclusion) and some examples.
| Aug. 13 | RSA, solvability of polynomials by radicals and constructability in brief. None of this material will be tested on the final exam.
| Aug. 14 | Review Session 1 : We looked at some TRUE/FALSE questions and their SOLUTIONS. Aug. 15 | Review Session 2
| Aug. 16 | Final Exam 12:10 to 2:00 p.m. in 6 Evans. You will need only a pencil or pen and your considerable knowledge of abstract algebra! |