🧱 Foundational Study Roadmap for PrivNet.AI & Silverman

This roadmap outlines the mathematical prerequisites for studying The Arithmetic of Elliptic Curves (Silverman) and building a strong theoretical base for the PrivNet.AI project.

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📦 Phase 1 – Abstract Algebra Foundations

1. Sets, Maps, and Binary Operations

   • How to Prove It – Daniel J. Velleman (2nd Ed., 2006)
   • Naive Set Theory – Paul Halmos (1960, Dover reprint 2017)
   • Book of Proof – Richard Hammack (3rd Ed., 2022 – Free online)

2. Groups and Subgroups

   • Abstract Algebra – Dummit & Foote (3rd Ed., 2004)
   • A First Course in Abstract Algebra – Fraleigh (7th Ed., 2002)
   • Algebra: Chapter 0 – Paolo Aluffi (2009)

3. Rings, Ideals, and Homomorphisms

   • Basic Abstract Algebra – Bhattacharya & Jain (2nd Ed., 1994)
   • A Book of Abstract Algebra – Charles C. Pinter (2nd Ed., 2010)
   • Introduction to Commutative Algebra – Atiyah & MacDonald (2016)

4. Fields and Integral Domains

   • Field and Galois Theory – Patrick Morandi (1996)
   • Contemporary Abstract Algebra – Gallian (10th Ed., 2021)
   • Algebra – Michael Artin (2nd Ed., 2010)

5. Quotient Structures and Isomorphism Theorems

   • Topics in Algebra – I. N. Herstein (2006)
   • Advanced Modern Algebra – Joseph J. Rotman (2nd Ed., 2010)
   • Elements of Modern Algebra – Linda Gilbert (8th Ed., 2013)

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📦 Phase 2 – Linear Algebra

1. Vector Spaces and Subspaces

   • Linear Algebra Done Right – Sheldon Axler (3rd Ed., 2015)
   • Linear Algebra and Learning from Data – Gilbert Strang (2019)
   • Introduction to Linear Algebra – Serge Lang (5th Ed., 2018)

2. Linear Maps and Matrices

   • Matrix Analysis – Horn & Johnson (2nd Ed., 2012)
   • Finite-Dimensional Vector Spaces – Paul Halmos (2017)
   • Linear Algebra – Hoffman & Kunze (2nd Ed., 2018)

3. Eigenvalues and Eigenvectors

   • Linear Algebra: Theory, Intuition, Code – Mike X Cohen (2021)
   • Applied Linear Algebra – Olver & Shakiban (2018)
   • Numerical Linear Algebra – Trefethen & Bau (2017)

4. Duality, Transpose, Inverse Matrix

   • Advanced Linear Algebra – Steven Roman (3rd Ed., 2010)
   • Linear Algebra and Its Applications – David C. Lay (5th Ed., 2020)
   • Linear Functional Analysis – Bryan Rynne (2017)

5. Inner Product, Norm, Inner Product Spaces

   • Functional Analysis – Kreyszig (2017)
   • Inner Product Spaces and Applications – T. S. Blyth (2005)
   • Linear Algebra with Applications – Gareth Williams (9th Ed., 2017)

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📦 Phase 3 – Polynomial Theory

1. Polynomials and Roots

   • Contemporary Abstract Algebra – Gallian (10th Ed., 2021)
   • Polynomials – E. R. Barbeau (2003)
   • A Course in Algebra – E. B. Vinberg (2016 edition)

2. Algebraically Closed Fields

   • Algebra – Michael Artin (2nd Ed., 2010)
   • Topics in Algebra – I. N. Herstein (2006)
   • Basic Algebra I – Nathan Jacobson (2009)

3. Factorization and Inequalities

   • Abstract Algebra – Dummit & Foote (3rd Ed., 2004)
   • Number Theory: A Classical Introduction – Ireland & Rosen (2005)
   • Understanding Analysis – Stephen Abbott (2nd Ed., 2015)

4. Division Algorithms and Remainder Theorem

   • Introduction to Commutative Algebra – Atiyah & MacDonald (2016)
   • Polynomial Algebra – Lidl & Pilz (1998)
   • Field and Galois Theory – Patrick Morandi (1996)

5. Fundamental Theorem of Algebra

   • Visual Complex Analysis – Tristan Needham (1997, reprint 2021)
   • Complex Made Simple – David C. Ullrich (2008)
   • Principles of Mathematical Analysis – Walter Rudin (3rd Ed., 1976)

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📦 Phase 4 – Introductory Galois Theory

1. Field Extensions

   • Galois Theory – Ian Stewart (4th Ed., 2015)
   • A Course in Galois Theory – D. J. H. Garling (2009)
   • Lectures on Field Theory and Galois Theory – Steven H. Weintraub (2017)

2. Algebraic and Separable Extensions

   • Algebra – Serge Lang (2005)
   • Field Theory – Steven Roman (2006)
   • Abstract Algebra: Theory and Applications – Judson (2023)

3. Automorphisms and Galois Groups

   • Modern Algebra – John R. Durbin (7th Ed., 2018)
   • Field and Galois Theory – Patrick Morandi (1996)
   • Intro to Abstract Algebra – Nicholson (4th Ed., 2012)

4. Fundamental Theorem of Galois Theory

   • Galois Theory through Exercises – Juliusz Brzeziński (2020)
   • A Guide to Galois Theory – Helmut Völklein (2021)
   • Galois Theory for Beginners – Andre L. Yandl (Open Source)

5. Classic Examples (e.g. ℚ(√2), ℚ(ζₙ))

   • Field Theory and Its Classical Problems – Hadlock (2000)
   • Algebraic Equations – Siegel (2009)
   • Constructibility and Galois Theory – M. Aigner (2018)

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📦 Phase 5 – Foundational Algebraic Geometry

1. Affine Space ( \mathbb{A}^n )

   • Ideals, Varieties, and Algorithms – Cox, Little, O'Shea (4th Ed., 2015)
   • Undergraduate Algebraic Geometry – Miles Reid (2013)
   • Basic Algebraic Geometry I – Shafarevich (3rd Ed., 2013)

2. Zero Sets of Polynomials

   • Algebraic Curves – Fulton (2008)
   • Using Algebraic Geometry – Cox, Little, O'Shea (2nd Ed., 2005)
   • An Invitation to Algebraic Geometry – Smith et al. (2000)

3. Ideals and Varieties

   • Commutative Algebra – Eisenbud (1995)
   • Intro to Algebraic Geometry – Brendan Hassett (2007)
   • Ideals and Reality – Stillman & Eisenbud (2020)

4. K-Rational Points and Algebraic Closure

   • The Arithmetic of Elliptic Curves – Joseph Silverman (2nd Ed., 2009)
   • Elliptic Curves – Lawrence Washington (2008)
   • Rational Points on Varieties – Bjorn Poonen (2017)

5. Simplified Nullstellensatz Theorem

   • Algebraic Geometry – Robin Hartshorne (1977)
   • Real Algebraic Geometry – Bochnak et al. (1998)
   • From Polynomials to Sums of Squares – Bruce Reznick (2022)