# Principal Ideal Domains non-example

A good practice in math, especially when studying a type of mathematical objects, is by considering what doesn’t fit the definition.

I took Graduate Algebra with Professor Matthias Beck in Spring 2017. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Polynomial rings,
• irreducibility criteria,
• Gröbner bases & Buchberger’s algorithm,
• field extensions,
• splitting fields,
• Galois groups,
• fundamental theorem of Galois theory,
• applications of Galois extensions,
• introduction to the polynomial method with applications in graph theory and incidence geometry.

# Notes: Modern Algebra II

I took Modern Algebra II with Professor Matthias Beck in Spring 2016. These comprehensive notes were compiled using lecture notes and the textbook, David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Featured Image: Icosahedron and dodecahedron Duality
Credit: Images from Algebra: Abstract and Concrete by Frederick M. Goodman

Disclaimer: my notes are meant to be a toolbox while doing proofs and studying/practicing the course in general. There may contain typos or mistakes. Please feel free to let me know if you find any errors!

Topics Covered:

• Review of basic properties of groups and rings and their quotient structures and homomorphisms,
• group actions,
• Sylow’s theorems,
• principal ideal domains,
• unique factorization,
• Euclidean domains,
• polynomial rings,
• modules,
• field extensions,
• primitive roots,
• finite fields.

# Notes: Number Theory and Modern Algebra I

I took Modern Algebra I with Professor Matthias Beck in Fall 2015. These comprehensive notes were compiled using lecture notes and the textbooks,