1.2 Polynomial rings and irreducible polynomials

Polynomial rings are the backbone of field theory, forming a crucial link between abstract algebra and concrete equations. They provide a structured way to work with polynomials, offering tools to analyze their properties and relationships.

Irreducible polynomials play a starring role in this mathematical drama. Like prime numbers in integer arithmetic, they're the building blocks of polynomial factorization. Understanding them is key to solving complex equations and constructing new fields.

Polynomial rings and properties

Definition and notation

• Polynomial ring denoted as $R[x]$ set of all polynomials with coefficients from a ring $R$
• Coefficients come from a ring $R$ (integers, real numbers, complex numbers)
• Examples of polynomial rings: $\mathbb{Z}[x]$ (integers), $\mathbb{R}[x]$ (real numbers), $\mathbb{C}[x]$ (complex numbers)

Algebraic properties

• Polynomial rings are commutative rings satisfy commutative property of addition and multiplication
  • $f(x) + g(x) = g(x) + f(x)$
  • $f(x) \cdot g(x) = g(x) \cdot f(x)$
• Polynomial rings have a unity element constant polynomial $1$
• Polynomial rings are integral domains if the coefficient ring $R$ is an integral domain
  • No zero divisors: if $f(x) \cdot g(x) = 0$, then either $f(x) = 0$ or $g(x) = 0$
  • Example: $\mathbb{Z}[x]$ is an integral domain, but $\mathbb{Z}_6[x]$ (integers modulo 6) is not

Degree and leading coefficient

• Degree of a polynomial highest power of the variable in the polynomial
  • Example: $f(x) = 3x^4 + 2x^2 - 5$ has degree 4
• Leading coefficient coefficient of the highest degree term
  • Example: In $f(x) = 3x^4 + 2x^2 - 5$, the leading coefficient is 3
• Zero polynomial has degree $-\infty$ by convention
• Constant polynomials have degree 0

Irreducible polynomials

Definition and properties

• Irreducible polynomial cannot be factored into the product of two non-constant polynomials over a given field
• Analogous to prime numbers in the integers
• Example: $x^2 + 1$ is irreducible over $\mathbb{R}$, but reducible over $\mathbb{C}$ as $(x + i)(x - i)$
• Irreducibility depends on the field being considered

Irreducibility over specific fields

• Over the field of real numbers $\mathbb{R}$, a polynomial is irreducible if and only if it is:
  • Linear (degree 1)
  • Quadratic (degree 2) with a negative discriminant ($b^2 - 4ac < 0$)
• Over the field of complex numbers $\mathbb{C}$, every polynomial of degree greater than 0 is reducible
  • Fundamental Theorem of Algebra: every non-constant polynomial has a root in $\mathbb{C}$
• Eisenstein's criterion sufficient condition for a polynomial to be irreducible over the field of rational numbers $\mathbb{Q}$
  • If $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ with integer coefficients, and there exists a prime $p$ such that:
    • $p$ divides $a_0, a_1, \ldots, a_{n-1}$
    • $p$ does not divide $a_n$
    • $p^2$ does not divide $a_0$
  • Then $f(x)$ is irreducible over $\mathbb{Q}$
  • Example: $f(x) = x^3 + 3x + 6$ is irreducible over $\mathbb{Q}$ by Eisenstein's criterion with $p = 3$
• The polynomial $x^p - x$ is irreducible over the field of integers modulo $p$, where $p$ is a prime number
  • Used in the construction of finite fields

Polynomial arithmetic

Addition and subtraction

• Polynomial addition performed by adding the coefficients of like terms
  • Example: $(3x^2 + 2x - 1) + (2x^2 - 3x + 4) = 5x^2 - x + 3$
• Polynomial subtraction performed by subtracting the coefficients of like terms
  • Example: $(3x^2 + 2x - 1) - (2x^2 - 3x + 4) = x^2 + 5x - 5$
• Degree of the sum or difference of two polynomials is at most the maximum of the degrees of the individual polynomials

Multiplication

• Polynomial multiplication performed by multiplying each term of one polynomial by each term of the other polynomial and then adding the like terms
  • Example: $(3x + 2)(2x - 1) = 6x^2 + x - 2$
• Degree of the product of two polynomials is the sum of the degrees of the individual polynomials
  • $\deg(f(x) \cdot g(x)) = \deg(f(x)) + \deg(g(x))$
• Leading coefficient of the product of two polynomials is the product of the leading coefficients of the individual polynomials
  • $LC(f(x) \cdot g(x)) = LC(f(x)) \cdot LC(g(x))$
• Multiplication of polynomials is commutative, associative, and distributive over addition

Polynomial division and GCD

Division algorithm

• Division algorithm for polynomials: given two polynomials $f(x)$ and $g(x)$ with $g(x) \neq 0$, there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that:
  • $f(x) = g(x)q(x) + r(x)$
  • $\deg(r(x)) < \deg(g(x))$
• Example: Dividing $f(x) = x^3 + 2x^2 - 3x + 1$ by $g(x) = x^2 + 1$ gives:
  • $q(x) = x + 1$ and $r(x) = x + 2$
  • $f(x) = (x^2 + 1)(x + 1) + (x + 2)$
• Division algorithm is the basis for the Euclidean algorithm for finding the GCD of two polynomials

Greatest common divisor (GCD)

• GCD of two polynomials polynomial of the highest degree that divides both polynomials without a remainder
• GCD of two polynomials can be found using the Euclidean algorithm involves repeated application of the division algorithm
  • Example: GCD of $f(x) = x^3 - 3x + 2$ and $g(x) = x^2 - 1$ is $x - 1$
• If the GCD of two polynomials is 1, the polynomials are called relatively prime or coprime
• Bézout's identity: if the GCD of two polynomials $f(x)$ and $g(x)$ is $d(x)$, then there exist polynomials $a(x)$ and $b(x)$ such that:
  • $a(x)f(x) + b(x)g(x) = d(x)$
  • Example: For $f(x) = x^3 - 3x + 2$ and $g(x) = x^2 - 1$ with $d(x) = x - 1$, we have:
    • $a(x) = -x$ and $b(x) = x^2 - x - 1$
    • $(-x)(x^3 - 3x + 2) + (x^2 - x - 1)(x^2 - 1) = x - 1$