Summary
This is intended as a very brief indication of the material covered in each lecture.
Lecture 33: Any two splitting fields are isomorphic. Splitting fields are Galois extensions.
Lecture 32: 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. Statement of the Main Theorem of Galois Theory.
Lecture 31: 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.
Lecture 30: Two finite fields of the same order are isomorphic.
Lecture 29: There exists a field of order p^r for any prime p and any r≥1.
Lecture 28: 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. Example of constructing field of order 8.
Lecture 27: Examples of non-constructible numbers: π, 2^(1/3), cos(π/9). Impossible classical constructions: squaring a circle, doubling a cube, trisecting an angle. Finite fields. All finite fields have prime power order.
Lecture 26: Definition of constructible points in the Euclidean plane. Constructible real numbers as a subfield of the reals. Constructible real numbers are algebraic of degree a power of 2.
Lecture 25: 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).
Lecture 24: Algebraic and transcendental elements in a field extension. 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.
Lecture 23: Example of calculating the JNF of a matrix. Began final topic: FIELD EXTENSIONS. Every polynomial f ∈ F[x] has a root in some field extension E of F.
Lecture 22: Derivation (using the structure theroem) of Jordan Normal Form of a matrix or linear transformation. Relationship with the minimal polynomial and characteristic polynomial.
Lecture 21: Examples of invariant factor decomposition and primary decomposition for abelian groups. Began discussion of Jordan Normal Form for a linear transformation T:V → V of a complex vector space V.
Lecture 20: Example calculation of the invariant factor decomposition of a module. Application of structure theorem to abelian groups.Primary decomposition.
Lecture 19: 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.
Lecture 18: Every matrix over a PID is equivalent to a diagonal matrix with each diagonal entry dividing the following (continued). Example calculation.
Lecture 17: Matrix representation of a homomorphism between free modules. Definition of equivalence of matrices. Began argument that every matrix over a PID is equivalent to a diagonal matrix
Lecture 16: The Splitting Lemma. Every submodule of a finitely generated module over a PID is itself free.
Lecture 15: For an integral domain R, if R^m and R^n are isomorphic as R-modules, then m=n. Every (finitely generated) module is a quotient of a free module. Torsion elements and torsion submodule T of an R-module M. If R is an ID then M/T is torsion-free.
Lecture 14: Definition of a free module. Examples. Alternative definition of basis in terms of homorphisms. If M has a finite basis then it is isomorphihc to R^n.
Lecture 13: MODULES: Definition and examples. Submodules, module homomorphisms, quotient modules, isomorphism theorems. Definition of a basis.
Lecture 12: 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 showed the a prime integer p>2 can be written as a sum of two squares only if p≡1 (mod 4).
Lecture 11: Every ED is a PID. Eisenstein's irreducibilty criterion.
Lecture 10: R a UFD ⇒ R[x] a UFD. Definition of Euclidean Domain. The Gaussian integers as an example of a ED.
Lecture 9: Definition of primitive polynomial. Gauss Lemma. Difference between irreducible in R[x] and irreducible in F[x] (where R is a UFD and F its field of quotients).
Lecture 8: Finished showing that all PIDs are UFDs. In a UFD, p irreducible ⇔ p prime. Definition of greatest common divisor in a UFD.
Lecture 7: (p) prime iff p is prime. (p) maximal ⇒ p irreducible. Converse holds in a PID. p prime ⇒ p irreducible. Converse holds in a PID. Definition of Unique Factorisation Domain. Ascending chain condition for PIDs. Started proof that all PIDs are UFDs.
Lecture 6: Z[x] is not a PID. Prime and maximal ideals. R/I an integral domain iff I prime. R/I a field iff I is maximal. (R being a commutative, unital ring.) Definition of prime and irreducible elements in a ring. Definition of associates in a ring.
Lecture 5: Definition of Principal Ideal Domain (PID). Division algorithm for F[x], where F is a field. F[x] is a PID.
Lecture 4: Third isomorphism theorem (R/J)/(K/J) ≅ R/K, correspondence theorem, standard contructions (direct sum, ring of polynomials, ring of matrices), ideal generated by a subset, principal ideals.
Lecture 3: Quotient ring. First isomorphism theorem for rings. Second isomorphism theorem: (S+J)/J ≅ S/(S∩J)
Lecture 2: Definition of integral domain. Every field is an ID, every finite ID is a field. Subrings, homomorphisms, ideals.
Lecture 1: RINGS: Definition of a ring. Some examples. Units, fields,