401CRS Algebra 4

CMI, Jan-Apr 2026, Manoj Kummini
Milne (n.d.), Lang (2002), Artin (2011), Aluffi (2009), Isaacs (2009)

Topics:

• module theory (mostly over commutative rings): Hom and tensor; projective and injective modules. Structure theorem of modules over PIDs. application to linear operators.
• Galois theory: splitting fields, field automorphisms, fixed fields, Galois extensions, normal and separable extensions, examples.

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Lecture notes

Notation

1. Bilinear maps are barred in diagrams.
2. Function composition, when written as $fg$, is left to right.
3. Maps induced by universal properties are dashed or dotted in diagrams.

Modules

Categorical notes on Modules
LEC ALG4 1 ✅ Every vector space over a field has a basis
LEC ALG4 2 ✅ Free modules
Smith normal form
LEC ALG4 3 ✅ Projective Modules, invariance of cardinality of basis
LEC ALG4 4 ✅ Noetherian Modules, Hilbert’s basis theorem
LEC ALG4 5 ✅ Generators and relations, decomposition of finitely generated modules into torsion and free components
LEC ALG4 6 ✅ Finitely generated torsion modules over PIDs
LEC ALG4 7 ✅ Finitely generated $p$-primary modules
LEC ALG4 9 ✅ Jordan block decomposition
LEC ALG4 10 More on JCF, Cayley-Hamilton
LEC ALG4 11 Rational canonical form

Tensor products

LEC ALG4 12 ✅ Properties of tensor products
LEC ALG4 13 Tensor algebras
LEC ALG4 14 Symmetric algebras
LEC ALG4 15 Exterior algebras

Galois Theory

LEC ALG4 16

Tutorials

TUT ALG4 1
TUT ALG4 2

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References

Aluffi, P. (2009). Algebra: Chapter 0. American Mathematical Society.

Artin, M. (2011). Algebra (2. ed). Pearson Education, Prentice Hall.

Isaacs, I. M. (2009). Algebra: A Graduate Course. American Mathematical Society. https://doi.org/10.1090/gsm/100

Lang, S. (2002). Algebra (Vol. 211). Springer New York. https://doi.org/10.1007/978-1-4613-0041-0

Milne, J. S. (n.d.). Fields and Galois Theory.