### Math Studies Algebra, Fall 2015 and Spring 2016

From symmetry groups to fields

© 2015 Laure Bukh

Used with permission

#### When:

Mondays, Wednesdays, Fridays 10:30 (Spring semester)Mondays, Wednesdays, Fridays 13:30 (Fall semester)

#### Where:

Porter Hall 22A (Spring semester)Wean Hall 8427 (Fall semester)

#### What:

Algebra is the art of changing the perspective. The change is mainly achieved through abstraction, which strips the irrelevant details and brings the important to the forefront. The extra generality also enables the connections between far-flung mathematical concepts.

The aim of this course is both to introduce the algebraic way of thinking, and to convey the basic language of algebra. That language is the language of groups, rings, modules, fields. We shall see, for example, how the group theory unifies such topics as integer arithmetic, tessellations, solubility of polynomial equations, and counting holes in a pretzel. We shall also learn and use some category theory, which is a higher-level abstraction that unifies different algebraic notions.

#### Resources:

The book for the course is *Abstract Algebra*, 3rd ed., by Dummit and Foote. A copy of the first edition is on reserve in the library.

For the representation theory part of the course, we are mostly following *Linear Representations of Finite Groups* by Serre. It is
available online via library subscription.

There is a very cute proof by Arnold that quintic equations admit no ‘parametric’ solution. The video of Dror Bar-Natan presenting this proof in CMU is available from Prof. Bar-Natan's website. In class we prove a stronger result, but we must work harder for it.

Not all topics that we cover are in the book, and some topics we will cover differently.

Links to additional resources will be posted as the course progresses.

#### More fun (Spring semester):

More fun can be had at my *office hours* on Mondays 11:30am–12:30pm and Thursdays 10:30–11:30am in Wean 6202 or 6th floor lounge. I am also available by appointment.

#### More fun (Fall semester):

More fun can be had at my *office hours* on Mondays 2:30–3:30pm and Thursdays 9:30–10:30am in Wean 6202 or 6th floor lounge. I am also available by appointment.

#### Course activities (Spring and Fall semesters):

Mastery of any subject requires practice. Hence, there will be regular homeworks. For your own good, you are strongly encouraged to do as much homework as possible individually. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. All the writing must be done individually. Failure to do so will be treated as cheating. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. The main advantage of a class over just reading a textbook is the ability to ask questions, propose ideas, and interact in other ways. In particular, discussions during the lectures are encouraged.

In the fall semester, the homework will count for 25% of the grade. During the semester there will be three in-class tests (on **September 25**, **October 21**, **December 4**).
Each test will count for 15% of the grade. A take-home final exam will count for 30%. No collaboration or use of external resources is allowed on tests or on the final.

In the spring semester, the homework will count for 25% of the grade. It will be assigned biweekly. During the semester there will be two in-class tests (on **February 19** and on **April 1**). Each test will count for 22.5% of the grade. A take-home final exam will count for 30%.

Homework must be submitted in LaTeX via e-mail. I want both the LaTeX file and the PDF that is produced from it. The filenames must be of the form `lastname_alg_homeworknumber.tex`
and `lastname_alg_homeworknumber.pdf` respectively. Pictures do not have to be typeset; a legible photograph of a hand-drawn picture is acceptable.

The homework must be submitted by 10:30am of the day it is due (in the fall semester, it was due 1:30pm of the day it is due). For each *minute* that it is late, the grade will be reduced by 10%.

#### Lectures (Fall and Spring semesters):

- August 31: Introduction. Groups. Associativity as a consequence of function composition. Examples of groups.
*SL*(_{2}**R**). Dihedral groups. - September 2: Generators for the dihedral groups. Normal form for the dihedral group. Group presentations (somewhat informal). Symmetric groups. Cyclic permutations. Homework #1
- September 4: Fields. Matrix groups
*GL*and_{n}*SL*. Subgroups. The Heisenberg group. Homomorphisms. Isomorphisms. Group actions._{n} - September 7:
*Labor Day* - September 9: Partially ordered sets. Lattice of subgroups of a group. Centralizers and center. Conjugation. Conjugation in the symmetric group. Normalizers. Normal subgroups. A non-trivial normal subgroup of the dihedral group. Homework #2
- September 11: Subgroup generated by a subset: top-down and bottom-top approaches. Cyclic groups. Uniqueness of the cyclic groups.
- September 14: Orders of elements in cyclic groups. Zorn's lemma. Axiom of choice. Well-ordering principle. Every finitely generated group contains a maximal subgroup.
- September 16: Every finitely generated group contains a maximal subgroup (part II). Quotient groups. Kernels. Cosets. Homework #3
- September 18: Cosets. Lagrange's theorem. Normal subgroups again. Normality is not transitive. Set multiplication. Quotient groups.
- September 21: Index of a subgroup. Index-2 subgroups are normal. Cardinality of the product of subgroups.
- September 23: First isomorphism theorem. Third isomorphism theorem. Simple groups. Alternating group. Homework #4
- September 25:
**Test** - September 28: Alternating group is generated by 3-cycles. Alternating group is simple.
- September 30: Alternating group is simple. The 15 puzzle. Groups actions, revisited. Permutation representations. Linear representations. Left and right regular representations. Orbits. Kernels and faithfulness. Homework #5
- October 2: Transitivity. Stabilizers. Stabilizers in an orbit. Core of a subgroup. Conjugacy classes. Subgroups of index
*p*. - October 5: Largest subgroup of
*A*isomorphic to a symmetric group. Sizes of orbits in general, and conjugacy classes in particular. (Finite)_{n}*p*-groups. - October 7: Cauchy's theorem. First Sylow's theorem. Homework #6
- October 9: Second and third Sylow's theorems. Groups of order
*pq*(abelian case). - October 12: Groups of order 12. Arbitrary direct products. Direct sums. Semidirect product.
- October 14: Examples of semidirect products. Action of semidirect products. Classification of trivial semidirect products. Homework #7
- October 16: Groups of order 12. Upper central series. Nilpotent groups. Lower central series.
- October 19: The relation between the upper and lower central series. Finite nilpotent groups.
- October 21:
**Test**Homework #8 - October 23:
*Mid-semester break* - October 26: Rings. Examples of rings.
*M*(_{n}*R*). Units. Zero divisors. - October 28: Integral domains. Polynomial rings. Power series. Laurent series. Group rings. Ring homomorphisms. Ideals. Homework #9
- October 30: Quotient rings. Principal ideals. Ideals, rings, fields generated by a set. Complex numbers as a quotient ring. Categories.
- November 2: Examples of categories. How to stop worrying and love the universes. Products. Uniqueness of products. Opposite category. Coproducts. Products and coproducts in the category of abelian groups.
- November 4: Products and coproducts in the category of groups (without proofs). Free products. Fundamental groups (without proofs). Isomorphism theorems for rings. Maximal ideals. Prime ideals. Homework #10
- November 6: Rings of fractions (localization). Universal property of the ring of fractions. Universal property of quotients.
- November 9: More about universal properties. Field of fractions. Chinese remainder theorem.
- November 11: Euclidean domains. Gaussian integers are an Euclidean domain.
- November 13: Principal ideal domains. Euclidean domains are principal ideal domains. Irreducible and prime elements. Unique factorization domains. PID are UFD (existence part). Homework #11
- November 16: PID are UFD (uniqueness part). Polynomials rings over UFD are UFD.
- November 18: Irreducible polynomials. Roots of polynomials of a field. Eisenstein's criterion.
- November 20: Newton polygons and irreducibility. Noetherian rings. Hilbert basis theorem. Homework #12
- November 23: Monomial orderings. Gröbner bases. The division algorithm.
- November 25:
*Thanksgiving (part I).* - November 27:
*Thanksgiving (part II).* - November 30: Monomial ideals. Ascending chain condition. Syzigies. Buchberger's criterion (part I).
*Guest lecture by James Cummings* - December 2: Buchberger's criterion (part II). Buchberger's algorithm. Minimal Gröbner bases.
- December 4:
**Test** - December 7: Reduced Gröbner bases. Uniqueness of reduced Gröbner bases. Elimination ideals. Radical of an ideal (no proofs). Computing elimination ideals via Gröbner bases.
- December 9: Computing intersection of ideals. Modules. Modules over
**F**[x] (part I). - December 11: Modules over
**F**[x] (part II).**Z**-modules. Generation of modules. Direct sum of modules. Finitely generated modules over PID (statement). - December 14–20:
**Take-home final exam.**Final exam rules

- January 11: Bimodules.
*R*-algebras. Module homomorphisms. Quotients of*R*-modules. Free modules (definition and the universal property). - January 13: Free modules (construction). Tensor products of modules (part I).
- January 15: Tensor products of modules (part II).
- January 18:
*Martin Luther King day*Homework #13 - January 20: Tensor products of modules (part III).
- January 22: Tensor product of homomorphisms. Getting dirty with tensor products: finite-dimensional vectors spaces, and matrices.
- January 25: Symmetric algebra. Exterior algebra. Homework #14
- January 27: Geometric meaning of the exterior algebra.
- January 29: Noetherian modules. Rank of a module. Rank of a free module.
- February 1: Submodules of finitely generated free modules over PID are free.
- February 3: Structure theorem for finitely generated modules over PID. Homework #15
- February 5: Structure theorem for finitely generated modules over PID (uniqueness). Rational canonical form (part I).
- February 8: Rational canonical form (part II). Cayley–Hamilton theorem. Jordan normal form. Algebraically closed field.
- February 10: Geometric view of complex numbers. Fundamental theorem of algebra (with proof). Characteristic of a field. Linear representations. Homework #16
- February 12: Linear representations matrix-way. Trivial representation. Regular representation. Examples of linear representations. Irreduciblity and incomposablity. Permutation representations as linear representations. Decomposition of a permutation representation. Averaging trick.
- February 15: Maschke's theorem. Degree of a representation. Unitary representations. Characters. Schur's lemma.
- February 17: Orthogonality relations. Uniquesness of decomposition in irreducible representations. Homework #17
- February 19:
**Test** - February 22: Decomposition of the regular representation. Class functions. Number of characters.
- February 24: Examples of character tables. Abelian subgroups.
*Guest lecture by James Cummings*Homework #18 - February 26: Tensor square. Algebraic integers are a ring.
*Guest lecture by James Cummings* - February 29: Degree of representation divides order of the group.
- March 2: Hurwitz's theorem on product of sums of squares (part I).
- March 4–11:
*Spring break. No class.* - March 14: Hurwitz's theorem on product of sums of squares (part II).
- March 16: Field extensions. Degree of an extension. Homework #19
- March 18: Splitting fields. Simple extensions. Algebraic extensions. Iterated finite extensions.
- March 21: Cyclotomic extensions. Straightedge and compass constructions. Constructing regular
*n*-gons. - March 23: Algebraically closed fields. Algebraic closure. Normal extension. Separable polynomials. Homework #20
- March 25: Derivatives. Construction of finite fields. Separable extensions.
- March 28: Field automorphisms. Automorphism groups. Fixed fields. Number of automorphisms of a splitting field.
- March 30: Galois extensions. Galois groups. Degree of a fixed field (part I). Homework #21
- April 1:
**Test** - April 4: Degree of a fixed field (part II). Galois correspondence.
- April 6: Main trick of Galois theory. Embeddings and automorphisms. Normal subgroups of a Galois group. Homework #22
- April 8: Finite fields. Cyclotomic polynomials. Galois closure.
- April 11: Characterization of finite simple extensions. Primitive element theorem. Solving polynomial equations in radicals (part I).
- April 13: Solving polynomial equations in radicals (part II). Homework #23
- April 15:
*Carnival. No class.* - April 18: Solving polynomial equations in radicals (part III). Affine space. Affine varieties.
- April 20: Regular functions and regular maps. Coordinate rings.
- April 22: Radical ideals. Hilbert's Nullstellensatz (statement). Extension problem. Extension theorem (statement).
- April 25: Resultants. First elimination ideal.
- April 27: Extension theorem (proof). Weak Nullstellensatz.
- April 29: Strong Nullstellensatz. Zariski topology.
- May 2–9:
**Take-home final exam.**Final exam rules