8000 Syllabus Fall 2006.
Instructor: Roy Smith
Office hours Boyd 448, MWF 11:15-noon, and by appt.
Contact: roy@math.uga.edu, rcsmith97@comcast.net, 706-542-2595.
Grades will be based on performance in class, on a mid term exam, a final exam and weekly homework.
Goals: To prepare students to use the basic tools of commutative and non commutative algebra, (groups, rings, fields, matrices), and pass the PhD alg. prelim.
Content of course:
I. Linear and commutative algebra
We will treat commutative algebra first, generalizing the theory of vector spaces. The fundamental concept is Òlinear combinationsÓ.
i) Abelian groups
First we treat abelian groups, representing them as cokernels of maps between free abelian groups, especially finitely generated ones using matrices. Elementary row and column operations let us diagonalize integer matrices and prove the fundamental structure theorem: all finitely generated abelian groups are isomorphic to products of cyclic groups.
ii) Commutative rings
We discuss rings more general than integers, making each important theorem about integers into a definition.
We study Òintegral domainsÓ, rings with no zero divisors, "Euclidean domains" which admit a Euclidean algorithm (polynomials in one variable over a field, Gaussian integers), Òp.i.d.'sÓ in which every ideal (subgroup closed under multiplication by R) has a single generator (e.g. local rings of smooth curves), "u.f.d.'s" which admit unique factorization into irreducibles (polynomials over a field or over any u.f.d.), ÒnoetherianÓ rings where every ideal has a finite set of generators, (any quotient ring of a polynomial ring over a field), Òintegrally closedÓ domains R for which every root of a monic polynomial which lies in the fraction field, already lies in R, (the coordinate ring of an affine hypersurface which is smooth in codimension one), and ÒDedekind domainsÓ: normal domains where every proper prime ideal is maximal, (e.g. the affine ring of a smooth curve, or the ring of integers in a number field).
iii) R modules
Analogous to multiplying vectors by scalars from a field, we define ÒR modulesÓ as abelian groups which allow multiplication by elements of a ring R. For rings R which share those properties of Z used in the proof of the fundamental structure theorem, we obtain analogous theorems for R modules. Every finitely generated module over a noetherian ring is the cokernel of a matrix. Every matrix over a Euclidean domain can be diagonalized by elementary row and column operations, and by invertible secondary operations over a p.i.d. Hence we get structure theorems for finitely generated modules over p.i.d.Õs, and an algorithm for computing the decomposition over Euclidean domains.
There are similar theorems for modules over Dedekind domains.
iv) Canonical forms of linear operators
Applications include the important case of finitely generated ÒtorsionÓ modules (analogous to finite abelian groups) over k[X] where k is a field, i.e. pairs (V,T) where V is a finite dimensional k - vector space, and T is a k linear transformation. The structure theorem gives rational and Jordan canonical forms for T, and diagonalization criteria for the matrix of T. The characteristic polynomial replaces the order of a finite group, and and the minimal polynomial replaces the annihilator. The Cayley Hamilton theorem follows.
This completes the first half of the course.
II. Non commutative algebra: groups and field extensions
The basic concept in non abelian group theory is ÒconjugationÓ, studying the extent to which the action sending y to a^(-1) y a is non trivial.
i) Groups, Existence of subgroups, Sylow, Subnormal towers
The first goal is to understand something about the elements and subgroups of a given group, just from knowing its order.
Defining a homomorphism out of G is most naturally accomplished by finding an ÒactionÓ of G on a set S, which yields a homomorphism of G into Sym(S) the group of bijections S--->S or ÒsymmetriesÓ of S.
These actions are used in the proof of the Sylow theorem: for every prime power p^r dividing #G, there exist subgroups of G of order p^r and elements of order p.
A non abelian group is characterized by the fact that some conjugation actions are non trivial. Letting G act on its Sylow subgroups by conjugation or translation, can provide non trivial homomorphisms and non trivial normal subgroups.
As a substitute for the product decomposition of a finite abelian group into cyclic groups, we have the concept of a ÒsimpleÓ group (having only trivial normal subgroups), and a ÒsubnormalÓ tower for G in terms of simple constituents which are uniquely determined by G (Jordan Holder).
ii) Free groups,
Free abelian groups, from which homomorphisms to abelian groups are easy to define, must be replaced by Òfree [non abelian] groupsÓ, which allow easy homomorphisms to all groups, but whose structure is much harder to understand. Thus although every finite group G is the quotient of a free group by a (free) subgroup, this information is harder to use as the presentation of G by a map between free groups is harder to simplify, since matrices are inapplicable. M. ArtinÕs discussion of the Todd - Coxeter algorithm is apparently relevant here.
iii) Semi direct products
To classify even small non abelian groups, we need some standard examples and some standard constructions. Basic examples include symmetric groups and dihedral groups. Most non abelian groups do not decompose as direct products, but we learn to recognize those which do. We show that many small groups do decompose as Òsemi directÓ product of subgroups, and learn to recognize when a group has such a decomposition. Semi direct products of abelian groups can be non abelian. Dihedral groups D(2n) are semi direct products of the two cyclic groups, Z/2 and Z/n.
iv) Galois groups of field extensions
We examine Galois' use of group theory to decide when a polynomial with roots in an extension of k, can be expressed in terms of Ònth rootsÓ and field operations. The answer is given by analyzing a subnormal tower of the subgroup G of those field automorphisms of the full root field, which leave the coefficient field pointwise fixed. Solvability of the polynomial is equivalent (over Q) to a subnormal tower for G having only abelian simple constituents, and this is necessary for solvability over any field.
As a technically convenient device, we construct a universal algebraic extension of a given base field, its Òalgebraic closureÓ, using ZornÕs lemma.
v) Examples, computations of Galois groups
We compute a few small Galois groups, including some of form D(2n), S(n), and recall the structure of subnormal tower for these groups, relating it to solvability of polynomials. We deduce AbelÕs theorem that a general, polynomial of degree 5 or more is not solvable by radicals. Then we discuss fields associated to cyclotomic polynomials X^n - 1, whose Galois groups are abelian.
This description of the course is subject to variation.