Good algebra book to cover these topics?

manpreet Best Answer 2 years ago

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your advice on this, so please don't delete this topic like some other topics I made (please?)

The topics we will cover in the first module (called "further linear algebra") are:

1. Revision of Linear Algebra, change of bases, characteristic polynomial, eigenvalues, traces.

2. Hamilton-Cayley theorem, minimal polynomial, nilpotent matrices, Jordan forms of nilpotent matrices, primary decomposition.

3. Jordan normal form, functions of matrices, norms and convergence.

4. Powers and exponents, applications to difference and differential equations.

5. Bilinear forms, Quadratic forms, GLn- classification.

6. Orthogonal matrices, Gramm-Schmidt process, O(n)-classification, normal matrices.

7. Spectral theorem and applications to hypersurfaces, classifications of quadrics in R3 .

8. Hermitian, unitary and normal matrices, applications to quantum mechanics.

9. Linear algebra over Z, abelian groups, GLn(Z) and SLn(Z), examples of quadratic forms over Z.

10. Echelon forms of matrices over Z, application to finitely generated abelian groups.

And for the second algebra module ("groups and rings"):

1. Groups and subgroups, examples, rings and subrings, the issue of 1 in signature, fields, examples.

2. Isomorphisms, cyclic groups, orders of an element in a group, generators and relations, Cosets and Lagranges theorem, Eulers theorem, RSA, groups of order 4 and p.

3. Homomorphisms, examples of homomorphisms, kernels and images, normal subgroups and ideals, classification of groups of order 6 and 8.

4. Quotient groups and quotient rings, examples, isomorphism theorems, Cayley Theorem, Chinese remainder theorem.

5. Group actions on sets, groups, rings, graphs etc., orbit-stabilizer theorem, conjugacy classes, conjugacy classes in Dn, Sn and GLn, simplicity of A5.

6. Class equation, groups of order p 2 , semidirect products, classification of groups of order 2p.

7. Sylows theorems, classification of groups of order 12 and 15.

8. Polynomial rings, Euclidian domains, PID-s, UFD-s, rings of integers.

9. Gauss theorem about polynomial rings, matrix rings.

10. Ideals in matrix rings, idempotents, cyclotomic polynomials, discrete Fourier transform.

I am thinking that Artin is a good choice of book, since I have already covered the basics of groups, rings and linear algebra last year but I am more looking for a book that will cover the topics above most comprehensively (but not an encyclopedic one such as dummit/foote since I won't have too much time mainly due to work) in preference to a "fun" book if that makes sense (but of course if there is a fun book that covers most of the topics I would love to hear about it, especially if it is Artin as I read some and like it so far)

Thanks!