1.3 Roots and factorization of polynomials

Roots and factorization are key to understanding polynomials over fields. They help us break down complex equations into simpler parts, revealing their structure and solutions. This knowledge is crucial for solving polynomial equations and exploring field extensions.

Unique factorization of polynomials mirrors prime factorization of integers. It allows us to express polynomials as products of irreducible factors, providing insights into their properties and roots. This concept forms the foundation for more advanced topics in field theory.

Polynomial Roots over Fields

Finding Roots and Their Multiplicities

• A root or zero of a polynomial $p(x)$ is a value $r$ such that $p(r) = 0$
• The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has at least one complex root
  • For example, the polynomial $x^2 + 1$ has no real roots, but it has the complex roots $i$ and $-i$
• Over the real numbers, a polynomial of odd degree always has at least one real root, while a polynomial of even degree may have no real roots
  • The polynomial $x^3 - 1$ has the real root $1$ and the complex roots $-\frac{1}{2} \pm \frac{\sqrt{3}}{2}i$
• The multiplicity of a root is the number of times it appears as a factor in the polynomial
  • A root with multiplicity 1 is called a simple root
  • For instance, in the polynomial $(x - 1)^2(x + 1)$, the root $1$ has multiplicity 2, and the root $-1$ has multiplicity 1

Properties and Techniques for Finding Roots

• The sum of the multiplicities of all roots of a polynomial is equal to the degree of the polynomial
  • A polynomial of degree $n$ has exactly $n$ roots, counting multiplicities
• Descartes' rule of signs can be used to determine the possible number of positive and negative real roots of a polynomial
  • The number of positive real roots is either equal to the number of sign changes between consecutive nonzero coefficients or is less than it by an even number
  • The number of negative real roots is the number of sign changes of $f(-x)$ or is less than it by an even number

Factoring Polynomials over Fields

Factoring over Different Fields

• Factoring a polynomial involves expressing it as a product of irreducible polynomials, which cannot be factored further
• Over the real numbers, a polynomial can be factored into a product of linear factors (corresponding to real roots) and irreducible quadratic factors (corresponding to complex conjugate pairs of roots)
  • The polynomial $x^3 - 1$ can be factored as $(x - 1)(x^2 + x + 1)$ over the real numbers
• Over the complex numbers, the fundamental theorem of algebra guarantees that every polynomial can be factored into a product of linear factors
  • The polynomial $x^3 - 1$ can be factored as $(x - 1)(x - \omega)(x - \omega^2)$ over the complex numbers, where $\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$ is a cube root of unity
• Over finite fields, polynomials can be factored using techniques such as the Berlekamp algorithm or Cantor-Zassenhaus algorithm

Irreducibility Criteria

• Eisenstein's criterion can be used to determine the irreducibility of a polynomial with integer coefficients over the rational numbers
  • If a polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ with integer coefficients satisfies:
    • $p$ divides each $a_i$ for $i = 0, 1, \ldots, n-1$,
    • $p$ does not divide $a_n$, and
    • $p^2$ does not divide $a_0$,
  • then $f(x)$ is irreducible over the rational numbers
  • For example, the polynomial $x^3 + 3x + 3$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion with $p = 3$

Euclidean Algorithm for Polynomials

The Euclidean Algorithm and Its Applications

• The Euclidean algorithm is an efficient method for finding the greatest common divisor (GCD) of two polynomials
• The algorithm involves repeatedly dividing the polynomials and replacing the divisor with the remainder until the remainder is zero
  • The last non-zero remainder is the GCD
• The extended Euclidean algorithm can be used to find the coefficients of a linear combination of the polynomials that equals their GCD (Bézout's identity)
  • For polynomials $f(x)$ and $g(x)$, the extended Euclidean algorithm finds polynomials $s(x)$ and $t(x)$ such that $s(x)f(x) + t(x)g(x) = \gcd(f(x), g(x))$

Properties of the GCD

• The GCD of two polynomials is unique up to multiplication by a non-zero constant
• If the GCD of two polynomials is 1, the polynomials are said to be relatively prime or coprime
  • For example, the polynomials $x^2 + 1$ and $x^3 - 1$ are coprime, as their GCD is 1

Unique Factorization of Polynomials

The Unique Factorization Theorem

• The unique factorization theorem states that every non-zero polynomial over a field can be written as a product of irreducible polynomials in a unique way, up to the order of the factors and multiplication by non-zero constants
• Irreducible polynomials over a field are the analogues of prime numbers in the ring of integers
  • An irreducible polynomial cannot be factored into a product of two polynomials of lower degree over the same field
  • For example, over the real numbers, the polynomial $x^2 + 1$ is irreducible, while over the complex numbers, it factors as $(x + i)(x - i)$

Applications and Implications

• The unique factorization theorem allows for the development of a theory of divisibility for polynomials over fields, similar to the theory of divisibility for integers
  • For polynomials $f(x)$ and $g(x)$, we say $f(x)$ divides $g(x)$ if there exists a polynomial $q(x)$ such that $g(x) = q(x)f(x)$
• The theorem is essential for understanding the structure of polynomial rings and their ideals, which play a crucial role in algebraic geometry and number theory
  • Polynomial rings over fields have many properties similar to the ring of integers, such as the existence of a division algorithm and the ability to perform polynomial long division