Theory of Equations – MIT‑Level Notes
(Mathematics – IIT‑JEE Main + Advanced)
Author: Expert Teacher
Audience: University students (MIT‑level) who already know elementary algebra, calculus, and basic abstract algebra.
Goal: Understand the deep structure of polynomial equations: from first principles to the frontier of algebraic geometry and Galois theory.
Table of Contents
| # | Section | Highlights |
|---|---|---|
| 1 | Introduction & Notation | Why study equations? |
| 2 | Fundamental Theorem of Algebra (FTA) | Existence, uniqueness, and complex roots |
| 3 | Symmetric Polynomials | Elementary symmetric sums, Newton’s identities |
| 4 | Vieta’s Formula | Proof via symmetric functions |
| 5 | Resultant & Discriminant | Eliminating variables, detecting multiple roots |
| 6 | Multiple Roots & Derivatives | Multiplicity, derivative test, power series |
| 7 | Sturm’s Theorem | Counting real roots, algorithmic |
| 8 | Descartes’ Rule of Signs | Sign changes → root bounds |
| 9 | Newton Sums | Recursive computation of power sums |
| 10 | Irreducibility & Field Extensions | Rational Root Test, Eisenstein, extensions |
| 11 | Galois Theory Basics | Solvability by radicals, Galois group |
| 12 | Edge Cases & Pathologies | Repeated roots, non‑real coefficients, infinite fields |
| 13 | Challenging Problem | A deep exercise with full solution |
| 14 | Further Reading | Textbooks and papers |
1. Introduction & Notation
- Polynomial over a field (F): [ P(x)=a_n x^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0,\qquad a_i\in F,;a_n\neq 0. ]
- Degree (n): the highest power of (x) with nonzero coefficient.
- Roots: elements (\alpha\in \overline{F}) (an algebraic closure of (F)) such that (P(\alpha)=0).
- Multiplicity: (\alpha) is a root of multiplicity (m) iff ((x-\alpha)^m) divides (P(x)) but ((x-\alpha)^{m+1}) does not.
- Field of coefficients: Usually (F=\mathbb{Q}), but we will also consider (\mathbb{R}), (\mathbb{C}), finite fields, and number fields.
2. Fundamental Theorem of Algebra (FTA)
Theorem (FTA).
Every non‑constant polynomial (P(x)\in\mathbb{C}[x]) of degree (n) has exactly (n) roots in (\mathbb{C}) (counted with multiplicity).
Proof Sketch (via Liouville)
- Assume (P) has no root. Then (1/P(x)) is an entire function.
- Show (|P(x)|\to\infty) as (|x|\to\infty) using the leading term: (|P(x)|\sim |a_n||x|^n).
- Therefore (|1/P(x)|\to 0) as (|x|\to\infty).
- Liouville’s theorem: a bounded entire function is constant → contradiction because (1/P(x)) is nonconstant.
- Thus (P) has a root; apply the same argument to (P/(x-\alpha)) to get all (n) roots.
Key Insight: Complex analysis guarantees existence; the count follows from multiplicity and factorization.
3. Symmetric Polynomials
Let the roots of (P(x)) be (\alpha_1,\dots,\alpha_n) (in (\overline{F})).
Define elementary symmetric sums:
[
\begin{aligned}
e_1 &= \sum_{i}\alpha_i, \
e_2 &= \sum_{i<j}\alpha_i\alpha_j, \
&\vdots \
e_n &= \alpha_1\alpha_2\cdots\alpha_n.
\end{aligned}
]
By the Fundamental Theorem of Symmetric Polynomials, any symmetric polynomial (S(\alpha_1,\dots,\alpha_n)) can be expressed uniquely as a polynomial in (e_1,\dots,e_n).
Example: (S=\sum_i \alpha_i^2 = e_1^2-2e_2).