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If you have questions or want to talk to me send me email at bergv@uiuc.edu. (In particular calling my office and leaving a message is not recommended if you want a reaction any time soon.) My official office hours are announced here. Extra homework office hours: Monday 3PM, in the elevator room, 3rd floor Illini Hall.
| Week # | Date | Topics covered | Homework Info | |
|---|---|---|---|---|
| Week 1 | ||||
| Tuesday 08/24 | Monoids, Groups, permutation groups, General linear group of a vector space, automorphism group of a group, categories, morphisms, automorphisms, examples: categories of sets, groups, vectorspaces. | |||
| Thursday 08/28 | Empty set, initial, terminal objects in a category, universal objects, commutative diagrams. Normal subgroups, kernels, cosets, quotient groups and universal mapping property; categorical formulation, cyclic groups, commutator subgroup. First Isomorphism theorem. | Homework 1:here. | ||
| Week 2 | ||||
| Tuesday 09/01 | Second/third Iso thm. Direct products, is $GL_2$ a direct product of $SL_2$ and $K^*$? | |||
| Thursday 09/03 | Exact sequences, split, simple groups, simple examples of nontrivial extensions. Group actions, orbits, isotropy group, stabilizer, index, orbit stabilizer theorem, orbit decomposition of a set with group action. Examples of actions. , | |||
| Week 3 | ||||
| Tuesday 09/08 | Conjugation action and class formula, centralizer, normalizer, fixed points, counting fixed points. Lagrange's thm. Cauchy's theorem, Fermat's little thm. Exponents, periods, sylow sub groups, p-groups have non trivial center. | Homework 2:here. | ||
| Thursday 09/10 | Sylow theorems. Simple examples. Unique Sylow groups are normal. Simplicity. Groups of order pq. | |||
| Week 4 | ||||
| Tuesday 09/15 | Normal towers of sub groups, solvable groups, p-proups are solvable; composition series, Schreier refinement, Jordan-Holder (no proofs). Abelian towers for a finite group has a cyclic refinement. Behavior of normal towers under homomorphisms. | Homework 3:here. | ||
| Thursday 09/17 | Dihedral groups, groups of order 2q, examples of non-split extensions. Solvable groups, commutator subgroup, derived series; symmetric group, cycles, cycle decomposition, cycle type, partitions, Young diagram. Classification of conjugacy classes in the symmetric group. | |||
| Week 5 | ||||
| Tuesday 09/22 | Number of conjugacy classes is number of partition. Action of the symmetric group on polynomials, the sign homomorphism, the alternating group. N is a normal subgroups iff it is a union of conjugacy classes. Conjugacy classes for An vs those of Sn: either equal or half size. Orbit split for An iff the cycle type has only distinct odd parts. | Homework 4:here. | ||
| Thursday 09/24 | The conjugacy classes of A5, and proof of simplicity of A5. 3-cycles generate An. A normal subgroup of An containing a 3-cycle contains them all, simplicity of An. Center of An is trivial. | |||
| Week 6 | ||||
| Tuesday 09/29 | 3-cycles are commutators, If N normal in G with Abelian quotient, then N contains the commutator subgroup. Sn is not solvable for n at least 5. Rings, rings without identity, left and right inverses, group of units of a ring, division rings, quaternion ring, ideals, homomorphisms of rings, kernels and quotients. | |||
| Thursday 10/1 | Principal ideals, isomorphism thm for rings. Zerodivisors, integral domains, fields, prime ideals, maximal ideals, polynomials, power series rings, units for polynomials and power series rings. Degree. Multiplicative sets, localization, ringss of quotients. Prime and irreducible elements. in UFD irreducibles are prime. PID. | Homework 5:here . | ||
| Week 7 | ||||
| Tuesday 10/6 | In a PID irreducibles are prime. Noetherian and Artinian rings. PIDs are Noetherian. PIDs are UFD. Euclidean functions and rings. | |||
| Thursday 10/08 | Exam I, on all the group theory material. | |||
| Week 8 | ||||
| Tuesday 10/13 | Z[x] not a PID. Greatest common divisors in Euclidean domains, PID, UFD. Order, content, primitive polynomials in D[x], F[x]. Gauss lemma (products of primitives is primitive). Reduction of polynomials modulo a prime. Theorem: id D is UFD then D[x] is UFD, and the primes in D[x] are the primes in D or the primitive irreducibles in F[x]. | Homework 6: here. | ||
| Thursday 10/15 | Irreducible polynomials over a field and maximal ideals. Homomorphisms of fields are embeddings. Field extensions F/E. Example: Q[\sqrt{-5}] is not a UFD. | |||
| Week 9 | ||||
| Tuesday 10/20 | Criteria for (ir)reducibilty: roots give factors, degree 2 and 3, reduction mod p, Eisenstein, cyclotomic polynomials, tricks to find roots of rational polynomials. Characteristic of a field. Prime fields. Trancendental/algebraic extensions. E[a] and E(a). Degree. Finite and finitely generated extensions. | Homework 7: here. | ||
| Thursday 10/22 | The minimal polynomial of a an algebraic element a in F/E. Extensions of embeddings and roots of minimal polynomials. Automorphisms. Algebraically closed fields. | |||
| Week 10 | ||||
| Tuesday 10/27 | There is a field extension containing a root of any irreducible f(x) in E[x]. There is a field F/E containing all roots of all f(x) in E[x]. Algebraic closure exists. | Homework 8: here. | ||
| Thursday 10/29 | Axiom of choice, Zorn's lemma (partially ordered sets, upperbounds for chains, maximal elements). Existence of maximal ideals, bases of vector spaces, and isomorphisms of algebraic closures., | |||
| Week 11 | ||||
| Tuesday 11/3 | Splitting fields, normal extensions, separable extensions. Derivative criterion for separability. | Homework 9: here. | ||
| Thursday 11/05 | Frobenius map. Existence and uniqueness of finite fields of order p^f. Simple field extensions, primitive elements. Automorphisms of F/E and degree, separability and normality. Galois extensions, Galois groups of extension, of polynomial. | |||
| Week 12 | ||||
| Tuesday 11/10 | Fundamental Thm of Galois theory. If F/E is Galois with
group G then F^G=E, hence injectivity of the Galois
correspondence. If F/E is finite and G| Homework
10: here. |
| ||
| Thursday 11/12 | Structure of cyclic groups and finite fields. Cyclic groups have a unique subgroup of each order dividing the order. Euler phi function, n=\sum_{n|d} \phi(d). If a finite group H has a bound d on the elements satisfying y^d=1, then H is cyclic. Every finite field has a cyclic multiplicative group, so all finite field extensions are simple (have a primitive element). | |||
| Week 13 | ||||
| Tuesday 11/17 | Separable degree, for simple extensions, multiplicativity of sep. degree. F/E finite seperable iff [F:E]_s=[F:E], all finite seperable extensions have a primitive element. Cyclotomic extensions, cyclotomic polynomials, Galois group of cyclotomic extensions. Symmetric polynomials: all symmetric polynomials are polynomials in the elementary symmetric polynomials. | |||
| Thursday 11/19 | EXAM II, | |||
| Week 14 | ||||
| Tuesday 11/24 | Thanksgiving Break. | |||
| Thursday 11/26 | Thanksgiving break, | |||
| Week 15 | ||||
| Tuesday 12/1 | . | Homework 12: | ||
| Thursday 12/03 | , | |||
| Week 16 | ||||
| Tuesday 12/8 | . | Homework 13: | ||
| Thursday 12/10 | Reading Day, | |||
| Friday 12/11 | Final, 8:00-11:00PM, in class | The final is cumulative, |