Algebra - Number Theory

                GROUP THEORY     &     RING THEORY

• Notes from the classroom(taught by professor Dustin Cartwright)

• Some problems with solutions(by Dimitrios Nikolakopoulos(me))

• Greek notes(taught by professor Dimitrios Chatzakos) Fall 2022

• Books: 

  1. 1. Abstract Algebra, D. S. Dummit and R. M. Foote

     2. An introduction to the theory of groups, J. J. Rotman

     3. Advanced Modern Algebra, J. J. Rotman

                MODULE THEORY     &     GALOIS/FIELD THEORY

• Notes from the classroom(taught by professor Shashikant Mulay)

• Notes from MIZZOU MATH Fall 2023 (taught by professor Dana Weston)

• Greek notes-GaloisTheory(by Dimitrios Nikolakopoulos(me)) Spring 2021

  1. 1. Θεωρία Galois, Θ.Θ. Αποστολίδη και Χ. Χαραλάμπους

     2. Πρόχειρες σημειώσεις Θεωρία Galois, Μιχάλης Μαλιάκας

     3. Εισαγωγή στη Θεωρία Galois, Γεώργιος Κασαπίδης

• Books: 

  1. 1. Abstract Algebra, D. S. Dummit and R. M. Foote

     2. Algebra, T. W. Hungerford

                         ARITHMETIC GEOMETRY

• Notes from the classroom(taught by professor Shashikant Mulay) Spring 2023

  1. 1. Existence and equicardinality of bases

     2. Flat-ness and Going Down Property

• Some problems with solutions(by Dimitrios Nikolakopoulos(me))

  1. 1. Has only finitely many points

     2. Factors of homogeneous polynomials are homogeneous

     3. Exercises with no solutions

• Greek notes(by Dimitrios Nikolakopoulos(me))

• Books: 

  1. 1. Algebraic Geometry and Arithmetic Curves, Q. Liu

     2. Algebraic Geometry, R. Hartshorne

     3. Algebraic Geometry - A First Course, J. Harris

     4. Arithmetic and Algebraic Geometry Notes, Shou-Wu Zhang

     5. Introduction to Arithmetic Geometry, Bjorn Poonen

                         ARITHMETIC GEOMETRY

• Notes from the classroom(taught by professor Shashikant Mulay) Spring 2023

• Some problems with solutions(by Dimitrios Nikolakopoulos(me))

  1. 1. Exercises with no solutions

• Greek notes(by Dimitrios Nikolakopoulos(me))

• Books: 

  1. 1. Algebraic Geometry, R. Hartshorne

     2. Algebraic Geometry - A First Course, J. Harris

     3. Algebraic Number Theory, A. Fröhlich and M. J. Taylor

                                      NUMBER THEORY

• Notes from the classroom(taught by professor Shashikant Mulay)

  1. 1. 1 mod 4 primes are sums of two integer squares

     2. Power Congruence

     3. Quadratic Reciprocity

     4. q-Jacobi Sums

     5. Sum of Mobius function over n

     6. Old notes from MATH UTK Fall 2021

• Some problems with solutions(by Dimitrios Nikolakopoulos(me))

• Notes from Purdue University Fall 2023 (taught by professor Trevor D. Wooley)

• Greek notes(by Dimitrios Nikolakopoulos(me))

  1. 1. Θεωρία Αριθμών(taught by professor Dimitrios Chatzakos) UPatras, Spring 2023

• Books: 

  1. 1. A Classical Introduction to Modern Number Theory, K. Ireland and M. Rosen

     2. Elementary Analytic and Geometric Number Theory, H. Iwaniec and W. Narkiewicz and J. Urbanowicz

     3. Geometric and Analytic Number Theory, E. Hlawka and J. Schoiβengeier and R. Taschner

                                      NUMBER THEORY

• Notes from the classroom(taught by professor Shashikant Mulay)

  1. 1. A Bernoulli number formula

     2. Addition formula for ℘

     3. An infinite product for sin z and formula for ζ(2r)

     4. Analytic functions defined by real integrals

     5. Arc-length integral for ellipse and cubic curves

     6. Dirichlet density

     7. Dirichlet Eta function

     8. Lattice moduli and complex-multiplication

     9. Periodic functions

• Some problems with solutions(by Dimitrios Nikolakopoulos(me))

• Books: 

  1. 1. A Classical Introduction to Modern Number Theory, K. Ireland and M. Rosen

     2. Analytic number theory, Henryk Iwaniec and Emmanuel Kowalski

     3. Multiplicative Number Theory, Harold Davenport

     4. Analytic Number Theory, Keith Conrad

     5. Bernoulli Numbers and Zeta Functions, Tsuneo Arakawa and Tomoyoshi Ibukiyama and Masanobu Kaneko

     6. L-συναρτήσεις χαρακτήρων Dirichlet και Θεωρία Πρώτων αριθμών, Διπλωματική εργασία-Λώλης Μιχαήλ

• Lecture notes

• Embedding dimension and Regularity

• Resultants and Discriminants

• Trace, Norm and Discriminant
  
  

  • Higher Topics

    1. Absolute irreducibility

    2. Algebraic Curves: some basic facts

    3. Bezout’s Theorem

    4. Generalized traces and complementary modules

    5. Differentials on curves

    6. Finite-ness theorems

    7. Hilbert-Samuel functions and polynomials

    8. Homogeneous localization

    9. Homogeneous coordinate rings

    10. Krull dimension

    11. Linear systems and Divisors on curves

    12. Parametrization

    13. Quotient fields of graded domains

    14. Theorems of Riemann-Roch, Clifford and Hurwitz