Class Log
References are to the textbook by Dummit and Foote.
Tuesday 3/10/20. Separable field extensions (13.5). Normal field extensions.
Thursday 3/5/20. Subfields of finite fields. The multiplicative group of a finite field is cyclic. Algebraically closed fields.
Tuesday 3/3/20. Adjoining roots. Splitting fields (13.4). Application: Finite fields. A finite field has order a power of a prime, and conversely for any prime power q=pn, there exists a finite field F of that order, and any two such are isomorphic. (F is a splitting field for xq-x over Z/pZ.)
Thursday 2/27/20. Field extensions. Algebraic and transcendental elements (13.2). The minimal polynomial of an algebraic element of a field extension. The degree [K:F] of a field extension F ⊂ K. For a tower of field extensions F ⊂ K ⊂ L we have [L:F]=[L:K][K:F]. Application: ruler and compass constructions (13.3).
Tuesday 2/25/20. The Dehn invariant of a polytope (an application of the tensor product). Fields (13.1). Examples. Characteristic of a field. The prime subfield.
Thursday 2/20/20. For all R-modules M the covariant functor (-)⊗R M is right exact. Restriction and extension of scalars.
Tuesday 2/18/20. No class (UMass Monday).
Thursday 2/13/20. Tensor product of R-modules (10.4 or Atiyah MacDonald pp. 24--31). Universal property of the tensor product. Examples.
Tuesday 2/11/20. Categories, functors, and natural transformations. Injective and projective R-modules. Split exact sequences.
Thursday 2/6/20. HomR(M,N). Examples. For all R-modules M, the contravariant functor HomR(-,M) and the covariant functor HomR(M,-) are left exact.
Tuesday 2/4/20. Exact sequences (10.5). The snake lemma.
Thursday 1/30/20. Computing the Jordan normal form of a square matrix. A finiteness lemma for modules: If R is a Noetherian ring, M is a finitely generated R-module, and N is a submodule of M, then N is finitely generated. (This lemma was used in MATH 611 in the proof of the structure theorem for finitely generated modules over a PID.)
Tuesday 1/28/20. The minimal polynomial and characteristic polynomial of a square matrix and their relation to the rational canonical form. A square matrix is diagonalizable over a field F iff all the roots of the minimal polynomial of A lie in F and there are no repeated roots. Computing the rational canonical form of a square matrix A with entries in a field F (using the Smith normal form of xI-A over F[x]).
Thursday 1/23/20. The rational canonical form of a square matrix (12.2). The Jordan normal form of a square matrix over an algebraically closed field (12.3).
Tuesday 1/21/20. Review of modules (10.1--10.3). The structure theorem for finitely generated modules over a PID (12.1).