Lecuture Outline
This is intended as an indication of the material to be covered each week.
Week 1: Definition of a ring and some basic examples. Elementary properties of rings. Cancellation law. Definition of integral domain, unit, field, subring. Homomorphisms, ideals, quotient rings, first isomorphism theorem.
Week 2: Second and third isomorphism theorems, correspondence theorem, standard contructions (direct sum, ring of polynomials, ring of matrices), subring (or ideal) generated by a subset. Definition of Principal Ideal Domain (PID). F[x] is a PID. Definition of prime and irreducible elements in a ring. Definition of associates in a ring. Prime and maximal ideals. R/I an integral domain iff I prime. R/I a field iff I is maximal. In a PID, (p) maximal iff p irreducible. Ascending chain condition for PIDs. Definition of Unique Factorisation Domain.
Week 3: Difference between irreducible in Z[x] and irreducible in Q[x]. Definition of primitive polynomial. Every PID is a UFD. Gauss Lemma. D a UFD ⇒ D[x] a UFD.
Week 4: Definition of Euclidean Domain. Examples, including Z[i]. Every ED is a PID. Eisenstein's irreducibilty criterion. Every polynomial f ∈ F[x] has a root in some field extension E of F. As an application of the concepts introduced so far, we show that a prime integer p>2 can be written as a sum of two squares only if p≡1 (mod 4). MODULES: Definition and examples.
Week 5: Submodules, module homomorphisms, quotient modules, isomorphism theorems. Definition of a free module. Alternative definition of basis of a free module. For an integral domain R, if R^m and R^n are isomorphic as R-modules, then m=n. Torsion elements and torsion submodule T of an R-module M. If R is an ID then M/T is torsion-free. The Splitting Lemma.
Week 6: Every submodule of a finitely generated module over a PID is itself free. Representation of a homomorphism between free modules by a matrix. Definition of equivalence of matrices. Every matrix over a PID is equivalent to a diagonal matrix with each diagonal entry dividing the following.
Week 7: Given a free module F, and a submodule N of F, there is a basis {f1,...,fn} of F and elements d1,...,dn of R such that d1|d2|...|dn and the non-zero elements of {d1f1,...,dnfn} form a basis for N. Structure theorem for finitely generated modules over a PID. Corollaries: (1) M=T⊕F where T is the torsion submodule of M and F is free; (2) if M is torsion-free then M is free. Application to abelian groups. Primary decomposition. Examples for abelian groups. Began discussion of Jordan Normal Form for a linear transformation T:V → V of a complex vector space V
Week 8: Derivation of Jordan Normal Form of a matrix or linear transformation. Relationship with the minimal polynomial. Example of calculating the JNF of a matrix. Relationship with the minimal polynomial. Cayley-Hamilton Theorem as a corollary of JNF. Uniquness of invariant factor decomposition. FIELD EXTENSIONS. Algebraic and transcendental elements in a field extension.
Week 9: Irreducible polynomial and degree of an algebraic element. If a is algebraic of degree n over F, then {1,a,...,a^(n-1)} is a basis for F(a) as a vector space over F. Algebraic extensions and finite extensions. If F ⊆ E ⊆ K are fields and both [K:E] and [E:F] are finite, then [K:F]=[K:E][E:F]. If a∈E is algebraic over F, and b∈ F(a), then deg(b,F) divides deg(a,F). Definition of constructible points in the Euclidean plane. Constructible real numbers as a subfield of the reals. Constructible numbers are algebraic of degree a power of 2.
Week 10: Impossible classical constructions: squaring a circle, doubling a cube, trisecting an angle. Finite fields. All finite fields have prime power order. If K is a field of order q=p^r, then all elements are roots of x^q-x. The multiplicative group K* is cyclic. Examples of constructing fields of order 4 and 8. There exists a field of order p^r. Two finite fields of the same order are isomorphic. F-automorphisms of an extension field K⊇F. Fixed subfield of a subgroup H<Aut(K). Splitting field of a polynomial. Definition of Galois group G(K/F) and of Galois extension.
Week 11: Example: Galois group of x^3+3x+1. If K is a splitting field of f ∈F[x], then any element of G(K/F) permutes the roots of f. Note: We consider only the case of fields of characteristic zero from now on. Any two splitting fields are isomorphic. Splitting fields are Galois extensions. Artin's Theorem: If G is a finite subgroup of Aut(K), then [K:K^G]=|G|. Corollary 1: For any finite extensions K of F, |G(K/F)| divides [K:F]. Corollary 2: If G is a finite subgroup of Aut(K), then K is a Galois extension of K^G with Galois group equal to G. Corollary 3: If K is a Galois extension of F, then the fixed field of G(K/L) is exactly F. Correspondence between intermediate fields and subgroups of the Galois group as in the main theorem of Galois theory.
Week 12: Existence of primitive elements for finite extensions. If G is a finite subgroup of Aut(K) and F is its fixed subfield, then any b ∈K is algebraic over F and its irreducible polynomial (over F) is (x-b1)(x-b2)...(x-bn) where {b1,...,bn} is the orbit of b under the action of G. Discussion of solution of polynomials by radicals. Revision.