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MT1 will cover the topics from the first 88 pages of Herstein, as well as the material we have covered in class and on the homework. 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. Midterm 1 and Solutions |
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MT2 will cover the topics from the first 171 pages of Hers
tein, as well as the material we have covered in class and on the homework.
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. 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 chapters 4 and 5, but you will not be tested on section 5.5. We covered sections 6.2,6.3,6.4. We also covered my notes on group actions. You should prepare for the final by: writing out and remembering the definitions of the terms we learned this semester, writing aout 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. 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 are the problems we looked at in the review session on Tuesday. And here are some solutions. Also check out the exams and solutions I posted as review material from the previous midterms. Final grades have been posted to Bearfacts. Thank you all for a great summer semester. I wish you all the best of luck for the fall. |
| June 26 | Sets, Mappings, and the A(S). We started talking about cycle notation for A({1,2,...,n})=S_n, and will finish up tomorrow. I handed out a course info sheet (with a terrible mistake on it: the correct version is posted above-- please note the date of the FINAL exam is the 17th of August.) I also handed out the first assignment, due Thursday in class.
Tonight, you should look over the three sections we covered (1.2,1.3,1.4), and skim through the section on Integers (1.5), which we will cover tomorrow. | |
| June 27 | We finished up talking about cycle notation and products and inverses of elements of S_n. Lots of information about the integers, divisibility, primes, gcd's and division algorithm. We spent the last 15 minutes talking about how to write the gcd (m,n) in the form am+bn. Tomight, look over the first two sections of chapter 2, and try to make a lot of headway on the assignment HW1. I announced that I will have additional office hours tomorrow from 11-12. I will also hold office hours tomorrw 2-3. Please stop by if you have questions about the first assignment. Check out the new "Homework help" link in the Homework section. | |
| June 28 | I announced a mistake in questions #9 from the first HW (see the HW section above. We talked about Z/nZ and then we talked about the definition of a group, and I gave a bunch of examples of abelian and non-abelian groups. We talked about how to prove a set G with an operation * is a group - namely by checking the axioms. HW1 is due tomorrow. | |
| June 29 | I changed the due date for HW2 to Wed. July 5th. But you should try to finish it before Monday anyway, so that you can chill out on July 4th! Today in class we talked about subgroups and we defined the cyclic subgroup generated by an element of a group. I collected HW1 and mentioned that solutions will be posted as soon as I get home this evening around 7:30. We spent the last half of the class talking about the new group U_n, otherwise known as (Z/nZ)^x. This weekend, you should read the first four sections of Chapter 2, which cover the material from today, yesterday and next Monday. | |
| July 3 | We defined equivalence relations, and talked about the two most important examples for us: the "mod n" equivalence on Z, and its generalization to the "mod H" equivalence on G where H is a subgroup of a the group G. We learned about the order of a group or a subgroup, and that the order of an element is the same as the order of the cyclic subgroup it generates. We talked about cosets, and in particular discovered that the cosets of H form a partition of G - this gave us a proof of Lagrange's Theorem that the order of a subgroup of a finite group divides the order of the group. You should be reading sections 2.5 to 2.7, which we will cover before the weekend, and which will be tested on Midterm 1 next Tuesday. Here are some notes I wrote a long time ago about equivalence relations (really just a long example). | |
| July 5 | We defined homomorphism, isomorphism, automorphism of groups, and discussed a few examples. We showed that the only cyclic groups are Z and Z/nZ (in the sense that any cyclic group is ismorphic to one of those). Then we started to look at the idea of "turning an arbitrary homomorphism into an isomorphism by "restrictig the target space to its image (that was the easy step to make it onto), and by INVENTING an equivalence relation on the domain (namely that two elements are equivalent iff they're mapped to the same place. We'll finish up this discussion tomorrow, so you should read over the sections on factor groups and on the homomorphism theorems. I collected HW2, and returned HW1, and announced that my office hours are 11-12 and 2-4 on Monday. | |
| July 6 |
We finished our discussion of Homomorphisms, which led us directly to the definitions of normal subgroups and factor groups. We developed the First Homomorphism Theorem from scratch, and wrote dow the Third one. The Second one is on your HW3. We also did an example in which we used the first homomorphism theorem to prove that a particular factor group G/N was isomorphic to R. (Here are some notes I wrote a long time ago about the First Homomorphism Theorem. And here are some notes about factor groups and the Third Homomorphism Theorem (in the notes, the theorem is called the `isomorphism theorem') | |
| July 10 | Review Session for MT1. Please bring lots of questions! I'll bring some too! | |
| July 11 | Midterm 1: 12:10-1:30 in our usual classroom. You will need only your pencil and your thinking-cap. | |
| July 12 | I handed back MT1. Here is some information about the way the class did. We talked about symmetries, and spent most of the first half of class talking about the dihedral groups D_n. In the second half of class, we talked about the symmetry groups of the platonic solids, and played with some shapes! | |
| July 13 | We talked about group actions. Here are some notes about what we saw in class on Thursday. 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. | |
| July 17 | We spent the class talking about a few applications of group actions, namely the proofs of Cayley's Theorem, and Cauchy's Theorem, and we talked about colorings of the tetrahedron, and the symmetry group of that polyhedron. The proofs I gave of the two theorems are essentially the same as those from your textbook, but in the language of group actions. | |
| July 18 | We covered the chapter on Conjugacy and Sylow Theorems. In particular, we defined the class equation, and we showed that groups of order p^n have non-trivial center, ans showed that groups of order p^2 are abelian. We did course evaluations (midsemester) and began the proof of Sylow's theorem We'll finish the last few details and state the other two sylow theorems tomorrow. | |
| July 19 | I went over the 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 defined the direct product of groups, and stated the Fundamental Theorem of Finite Abelian Groups. We did a bunch of examples about that Theorem. | |
| July 20 | 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 how we might want to ``turn a homomorphism into an isomorphism'' by the same type of construction we used for group homomorphisms, and saw that we needed to define ideal in order to force the quotient (i.e. the set of cosets) to be a ring. And we showed that the kernel of a ring is an ideal. | |
| July 25 | We continued our discussion of the `construction of an isomorphism from a homomorphism' and stated 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. 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. | |
| July 26 | 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 31 | Review Session | |
| Aug. 1 | Midterm 2 | |
| Aug. 2 | 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. Then we Looked at the complex numbers, and inparticular, 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. 3 | 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. I defined the word 'algebraic' and showed that algebraic elements over a field F have unique minimal monic polynomials which are irreducible. I reminded you that if we start with a field F, and some element a in an extension of F, that F[a] is by definition the smallest ring that contains all the elements from F as well as the element a. 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. I defined the same thing for F[x], F(x) where x is some transcendental element over F. For a transcendental element x over F, we shoed that F[x] is not isomorphic to F(x). And we ended with the statement of the theorem that for an algebraic element a over F, the ring F[a] is actually a field, and hence isomorphic to F(a), and that these are both isomorphic to F(x)/(p(x)) where p(x) is the (unique monic irreducible) minimal polynomial for a in F[x]. | |
| Aug. 7 | We proved the statement from the end of class the day before (see above). We saw that some consequences were that if p is the minimal polynomial for a and has degree n, then [F(a):F] = n. We saw as a corollary that if a and b are algebraic elements of an extension of F, then F[a] is isomorphic to F[b] iff a and b have the same minimal polynomial in F[x]. 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. 8 | 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. 9 | 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. 10 | Finite Fields (conclusion) and 'Constructibility'
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| Aug. 14 | RSA, some more fun stuff, and "What comes next after Math113". | |
| Aug. 15 | Review Session 1 : We looked at lots of TRUE/FALSE questions and their SOLUTIONS. | |
| Aug. 16 | Review Session 2 | |
| Aug. 17 | Final Exam 12:10 to 2:00 p.m. in 289 Cory. You will need only a pencil or pen and your considerable knowledge of abstract algebra! | |