Chapter 00: Polynomial Algebra¶

Prerequisites: Basic Number Systems (Real and Complex Numbers)

Chapter Outline: Definition of Polynomial Ring $F[x]$ → Division Algorithm → Euclidean Domain (ED) Properties → Greatest Common Divisor (GCD) and Bezout Identity → Factorization Theorem → Fundamental Theorem of Algebra → Discriminants and Resultants → Irreducibility Criteria (Eisenstein's Criterion) → Introduction to Matrix Polynomials and Minimal Polynomials

Extension: Polynomial theory is the algebraic foundation for understanding matrix eigenvalues, minimal polynomials, and the Jordan Canonical Form; Resultant theory is a cornerstone of elimination theory and algebraic geometry.

Polynomial algebra is not merely an extension of elementary algebra but the fundamental language of linear operator theory. In linear algebra, we are concerned not just with evaluating polynomials but with their behavior as algebraic objects—division, factorization, and their action on operator spaces. This chapter systematically establishes the algebraic structure of $F[x]$, providing a solid foundation for the subsequent study of spectral theory.

00.1 Structure of the Polynomial Ring¶

Definition 00.1 (Polynomial Ring)

Let $F$ be a field. The polynomial ring $F[x]$ is the set of all expressions of the form $f(x) = \sum_{i=0}^n a_i x^i$, where $a_i \in F$. - If $a_n \neq 0$, then $\deg f = n$ is the degree of $f$. - A polynomial with $a_n = 1$ is called a monic polynomial.

Theorem 00.1 (Division Algorithm)

For $f(x), g(x) \in F[x]$ with $g(x) \neq 0$, there exist unique $q(x), r(x) \in F[x]$ such that: $$f(x) = q(x)g(x) + r(x), \quad \deg r < \deg g \text{ or } r = 0$$ This demonstrates that $F[x]$ is a Euclidean Domain (ED).

Corollary 00.1 (PID and UFD Properties)

Since $F[x]$ is a Euclidean Domain, it is necessarily: 1. Principal Ideal Domain (PID): Every ideal in $F[x]$ can be generated by a single element. 2. Unique Factorization Domain (UFD): Every non-zero, non-unit polynomial can be uniquely factored into a product of irreducible polynomials.

00.2 GCD and the Bezout Identity¶

Definition 00.2 (Greatest Common Divisor)

$d(x)$ is the Greatest Common Divisor (GCD) of $f(x)$ and $g(x)$, denoted $\gcd(f, g)$, if $d$ divides both $f$ and $g$, and any other common factor divides $d$. By convention, the monic GCD is chosen.

Theorem 00.2 (Bezout's Identity)

Let $d(x) = \gcd(f, g)$. Then there exist polynomials $u(x), v(x) \in F[x]$ such that: $$u(x)f(x) + v(x)g(x) = d(x)$$ In particular, if $\gcd(f, g) = 1$, $f$ and $g$ are said to be coprime (or relatively prime).

00.3 Roots and Factorization¶

Theorem 00.3 (Remainder and Factor Theorems)

The remainder of $f(x)$ divided by $(x - a)$ is equal to $f(a)$.
$(x - a)$ divides $f(x)$ if and only if $f(a) = 0$.

Theorem 00.4 (Fundamental Theorem of Algebra)

Every complex polynomial of degree $n \geq 1$ has at least one root in $\mathbb{C}$. Consequently, any $f(z) \in \mathbb{C}[z]$ can be completely factored into $n$ linear terms: $$f(z) = c(z - \alpha_1)(z - \alpha_2) \cdots (z - \alpha_n)$$

00.4 Irreducibility Criteria¶

Theorem 00.5 (Eisenstein's Criterion)

Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \in \mathbb{Z}[x]$. If there exists a prime $p$ such that: 1. $p \nmid a_n$; 2. $p \mid a_i$ for all $i < n$; 3. $p^2 \nmid a_0$. Then $f(x)$ is irreducible over the field of rational numbers $\mathbb{Q}$.

00.5 Resultants and Discriminants¶

Definition 00.3 (Resultant)

The Resultant $\operatorname{Res}(f, g)$ of two polynomials $f$ and $g$ is the determinant of their Sylvester matrix. $\operatorname{Res}(f, g) = 0$ if and only if $f$ and $g$ share a common root in an algebraic closure.

Exercises¶

Solution

 Thus $q(x) = x^2 + 2$ and $r(x) = 0$. This shows $x^2+1$ is a factor.

Solution

 $x^2 - 1 = (x-1)(x+1)$.

The common factor is $(x-1)$, so $\gcd(f, g) = x-1$.

Solution

 Setting $x=a$ gives $f(a) = 0 + r$. If $f(a)=0$, then $r=0$, so $(x-a) \mid f(x)$.

Solution

 - Reducible over $\mathbb{C}[x]$ as it factors into $(x+i)(x-i)$.

Solution

Chapter Summary¶

This chapter establishes the foundation for operator algebra in linear algebra:

****: The algebraic structure of the polynomial ring (division, GCD) is highly analogous to the ring of integers, both being Euclidean Domains.

****: Established the correspondence between roots and linear factors; the Fundamental Theorem of Algebra guarantees the existence of eigenvalues for operators.

****: Extended scalar polynomials to matrix variables, and through tools like resultants, prepared for the study of polynomial eigenvalue problems.