Inverse Galois Problem-The Literal Version

Inverse Galois Problem - Literal Version: Find an Explicit Polynomial f_G(x) for Any Finite Group G Is every finite group the Galois group of a polynomial with rational coefficients? That is the 100-year-old Inverse Galois Problem (IGP). Most textbooks prove abstract existence. This book does it literally: Can you write down ONE explicit polynomial f_G(x) in Z[x] with Gal(f_G) = G? A polynomial you can type into PARI/GP and verify in 2 seconds with polgalois(f_G). The Literal Version makes IGP personal. Instead of "does G occur?", we ask "what is YOUR polynomial for G?" For C2 it's x^2-2. For A5 it's x^5+10x^3-10x^2+35x-18. For M11 it's degree 11. For M23? Nobody knows - you could be first. Inside this book: Part I - Solved Cases, Made Explicit: - Chapter 1: Abelian groups - Complete solution via Kronecker-Weber + Gaussian periods. Construct explicit polynomial for any C_n. - Chapter 2: Symmetric & Alternating Groups - Hilbert Irreducibility Theorem in action. Why S_n and A_n are easy. - Chapter 3: The Rigidity Method - The heavy weapon of Belyi, Fried, Matzat, Thompson. Explained without schemes, with 2 full worked examples: A5 and PSL(2,7). Part II - Why Some Groups Are Hard: - Chapter 4: Obstructions - Grunwald-Wang, Brauer embedding, Essential dimension. Why D4 is trivial but Quaternion Q8 needs 8-degree polynomial. Part III - Your Turn: - Chapter 5: Master Table - 25 verified literal polynomials from C2 to M11, each checked with PARI/GP - 7-Step Roadmap - How to find polynomial for YOUR group G using free tools: GAP, PARI/GP, SageMath, Magma - Appendix: Complete search scripts, Exercises with full solutions Who is this for: - Final year BSc / MSc Mathematics students who love explicit algebra - PhD students in Number Theory, Galois Theory, Computational Algebra - Researchers who want verifiable polynomials, not abstract theorems - Self-learners who can run polgalois(x^5-6*x+3) in PARI/GP No scheme theory required. Every polynomial is verifiable. Every search script is given. As of 2026, M23 and M24 remain open. Finding explicit degree 23 polynomial with Galois group M23 is the most famous open challenge in explicit IGP. This book gives you the tools to try.