Glossary

1.2 Polynomial rings and irreducible polynomials

4 min read•july 30, 2024

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 . Understanding them is key to solving complex equations and constructing new fields.

Polynomial rings and properties

Definition and notation

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

Algebraic properties

  • Polynomial rings are commutative rings satisfy commutative property of and
    • f(x)+g(x)=g(x)+f(x)f(x) + g(x) = g(x) + f(x)
    • f(x)⋅g(x)=g(x)⋅f(x)f(x) \cdot g(x) = g(x) \cdot f(x)
  • Polynomial rings have a unity element 11
  • Polynomial rings are integral domains if the coefficient ring RR is an
    • No zero divisors: if f(x)⋅g(x)=0f(x) \cdot g(x) = 0, then either f(x)=0f(x) = 0 or g(x)=0g(x) = 0
    • Example: Z[x]\mathbb{Z}[x] is an integral domain, but Z6[x]\mathbb{Z}_6[x] (integers modulo 6) is not

Degree and leading coefficient

  • of a polynomial highest power of the variable in the polynomial
    • Example: f(x)=3x4+2x2−5f(x) = 3x^4 + 2x^2 - 5 has degree 4
  • coefficient of the highest degree term
    • Example: In f(x)=3x4+2x2−5f(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

  • 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](https://www.fiveableKeyTerm:x^2_+_1) is irreducible over R\mathbb{R}, but reducible over C\mathbb{C} as (x+i)(x−i)(x + i)(x - i)
  • Irreducibility depends on the field being considered

Irreducibility over specific fields

  • Over the field of real numbers R\mathbb{R}, a polynomial is irreducible if and only if it is:
    • Linear (degree 1)
    • Quadratic (degree 2) with a negative discriminant (b2−4ac<0b^2 - 4ac < 0)
  • Over the field of complex numbers C\mathbb{C}, every polynomial of degree greater than 0 is reducible
    • Fundamental Theorem of Algebra: every non-constant polynomial has a in C\mathbb{C}
  • sufficient condition for a polynomial to be irreducible over the field of rational numbers Q\mathbb{Q}
    • If f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 with integer coefficients, and there exists a prime pp such that:
      • pp divides a0,a1,…,an−1a_0, a_1, \ldots, a_{n-1}
      • pp does not divide ana_n
      • p2p^2 does not divide a0a_0
    • Then f(x)f(x) is irreducible over Q\mathbb{Q}
    • Example: f(x)=x3+3x+6f(x) = x^3 + 3x + 6 is irreducible over Q\mathbb{Q} by Eisenstein's criterion with p=3p = 3
  • The polynomial xp−xx^p - x is irreducible over the field of integers modulo pp, where pp 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: (3x2+2x−1)+(2x2−3x+4)=5x2−x+3(3x^2 + 2x - 1) + (2x^2 - 3x + 4) = 5x^2 - x + 3
  • Polynomial subtraction performed by subtracting the coefficients of like terms
    • Example: (3x2+2x−1)−(2x2−3x+4)=x2+5x−5(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)=6x2+x−2(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)⋅g(x))=deg⁡(f(x))+deg⁡(g(x))\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)⋅g(x))=LC(f(x))⋅LC(g(x))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

  • algorithm for polynomials: given two polynomials f(x)f(x) and g(x)g(x) with g(x)≠0g(x) \neq 0, there exist unique polynomials q(x)q(x) (quotient) and r(x)r(x) (remainder) such that:
    • f(x)=g(x)q(x)+r(x)f(x) = g(x)q(x) + r(x)
    • deg⁡(r(x))<deg⁡(g(x))\deg(r(x)) < \deg(g(x))
  • Example: Dividing f(x)=x3+2x2−3x+1f(x) = x^3 + 2x^2 - 3x + 1 by g(x)=x2+1g(x) = x^2 + 1 gives:
    • q(x)=x+1q(x) = x + 1 and r(x)=x+2r(x) = x + 2
    • f(x)=(x2+1)(x+1)+(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)=x3−3x+2f(x) = x^3 - 3x + 2 and g(x)=x2−1g(x) = x^2 - 1 is x−1x - 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)f(x) and g(x)g(x) is d(x)d(x), then there exist polynomials a(x)a(x) and b(x)b(x) such that:
    • a(x)f(x)+b(x)g(x)=d(x)a(x)f(x) + b(x)g(x) = d(x)
    • Example: For f(x)=x3−3x+2f(x) = x^3 - 3x + 2 and g(x)=x2−1g(x) = x^2 - 1 with d(x)=x−1d(x) = x - 1, we have:
      • a(x)=−xa(x) = -x and b(x)=x2−x−1b(x) = x^2 - x - 1
      • (−x)(x3−3x+2)+(x2−x−1)(x2−1)=x−1(-x)(x^3 - 3x + 2) + (x^2 - x - 1)(x^2 - 1) = x - 1

Key Terms to Review (20)

Addition: Addition is a fundamental operation in mathematics that combines two or more numbers to produce a sum. This operation serves as a building block for more complex mathematical structures, providing the necessary framework for understanding concepts like fields, polynomials, and geometric constructs.
Algebraic Closure: An algebraic closure of a field is a field extension in which every non-constant polynomial has a root. It provides a comprehensive setting for understanding the solutions of polynomial equations and plays a crucial role in various mathematical areas, including Galois theory and number theory. In this context, it allows us to analyze the behavior of polynomials and their roots, connecting deeply with other important mathematical concepts.
Algebraic Extension: An algebraic extension is a type of field extension where every element of the extended field is algebraic over the base field, meaning each element is a root of some non-zero polynomial with coefficients in the base field. This concept plays a crucial role in understanding how fields can be expanded and how polynomials behave within those fields.
Constant polynomial: A constant polynomial is a polynomial of degree zero, which means it has the form $a$, where $a$ is a constant value and does not depend on any variable. This type of polynomial represents a horizontal line in the Cartesian plane and plays an important role in the study of polynomial rings and the classification of irreducible polynomials. Constant polynomials are integral in understanding the structure of polynomials and serve as the simplest case when analyzing polynomial properties.
Degree: In mathematics, particularly in algebra, the degree of a polynomial is the highest power of the variable in the polynomial expression. It serves as a fundamental concept that helps classify polynomials and understand their behavior, including roots and factorization. The degree is also crucial when discussing the impossibility of certain constructions and the nature of irreducible polynomials.
Division: In mathematics, division is the operation of determining how many times one number (the divisor) is contained within another number (the dividend). This concept is crucial in polynomial rings, where it refers to the process of dividing one polynomial by another to obtain a quotient and a remainder. The division algorithm for polynomials ensures that every polynomial can be expressed in terms of a quotient and a remainder, establishing a framework for understanding irreducible polynomials and their significance within the ring.
Eisenstein's Criterion: Eisenstein's Criterion is a powerful tool in algebra that provides a sufficient condition for determining the irreducibility of a polynomial with integer coefficients. Specifically, if a polynomial satisfies certain divisibility conditions with respect to a prime number, then it cannot be factored into lower-degree polynomials with integer coefficients. This criterion connects to the study of polynomial rings and the search for irreducible polynomials, as well as the analysis of roots and how polynomials can be factored over different fields.
Factorization: Factorization is the process of breaking down an object, such as a polynomial or a number, into a product of simpler factors that when multiplied together give back the original object. This concept is crucial in understanding polynomial rings and irreducible polynomials, as it helps identify the structure of polynomials and their roots. In the realm of Galois Theory, factorization plays a significant role in analyzing how the roots of polynomials relate to symmetry and group structures.
Field of fractions: A field of fractions is a construction that allows you to take an integral domain and create a field where every non-zero element of the integral domain can be expressed as a fraction. This concept is essential in understanding how to manipulate polynomials and their irreducibility, as it provides a way to consider ratios of polynomials, helping us analyze their roots and factorization properties more effectively.
Integral Domain: An integral domain is a type of commutative ring with no zero divisors and a multiplicative identity, which means that if the product of two non-zero elements is zero, then at least one of the elements must be zero. This property allows integral domains to support the cancellation of non-zero elements in equations, making them crucial in algebraic structures and number theory. Integral domains are essential for understanding concepts such as polynomial rings, irreducibility, and algebraic closures, since they provide a foundation for ensuring that division is well-defined when working with polynomials and their roots.
Irreducible Polynomial: An irreducible polynomial is a non-constant polynomial that cannot be factored into the product of two non-constant polynomials over a given field. This concept is crucial in understanding the structure of field extensions, as the irreducibility of a polynomial often determines the nature of roots and their relationships within these fields.
Leading coefficient: The leading coefficient is the coefficient of the term with the highest degree in a polynomial. This important characteristic determines the polynomial's end behavior and influences its overall shape when graphed. The leading coefficient plays a significant role in understanding the properties of polynomials, particularly when discussing factorization and the irreducibility of polynomials.
Monic Polynomial: A monic polynomial is a polynomial in which the leading coefficient (the coefficient of the highest degree term) is equal to 1. This characteristic makes monic polynomials particularly useful in algebra, especially when discussing factorization and irreducibility in polynomial rings.
Multiplication: Multiplication is an arithmetic operation that combines two elements to produce a third element, typically within a mathematical structure such as a field. It serves as a fundamental operation in various algebraic systems, where it adheres to specific properties like associativity, distributivity, and the existence of an identity element. This operation is crucial in understanding the structure of fields, especially in finite fields where multiplication helps to define their properties and behavior.
Polynomial Ring: A polynomial ring is a mathematical structure formed from the set of polynomials with coefficients in a given ring. This allows for operations like addition and multiplication of polynomials, leading to important concepts such as irreducibility and factorization, which play critical roles in understanding minimal polynomials and algebraic degrees, as well as the characteristics of algebraic closures.
Rational Root Theorem: The Rational Root Theorem is a tool used in algebra that provides a way to find all possible rational roots of a polynomial equation with integer coefficients. This theorem states that if a polynomial has a rational root expressed as $$\frac{p}{q}$$ (in lowest terms), then 'p' must be a factor of the constant term and 'q' must be a factor of the leading coefficient. This connection allows one to systematically identify potential rational roots, which is especially useful when dealing with minimal polynomials and understanding their algebraic degree, as well as analyzing irreducible polynomials within polynomial rings.
Root: In mathematics, a root of a polynomial is a value for which the polynomial evaluates to zero. This concept is essential in understanding polynomial equations, as finding roots allows for the factorization of polynomials and insights into their structure, including the construction of splitting fields and the nature of separable polynomials.
Symmetric polynomials: Symmetric polynomials are polynomials that remain unchanged when the variables are permuted. They play a crucial role in various areas of algebra, particularly in polynomial rings and the study of irreducible polynomials, as they help to understand the relationships between roots and coefficients.
X^2 + 1: The polynomial $x^2 + 1$ is a quadratic polynomial that is a fundamental example in algebra. It illustrates important concepts related to polynomial rings and the classification of irreducible polynomials. This polynomial cannot be factored over the real numbers, making it a key example in understanding irreducibility and the construction of extension fields.
X^3 - 2: The polynomial $x^3 - 2$ is a cubic polynomial that can be expressed in the form $f(x) = x^3 - 2$. This polynomial is significant because it has implications for the study of roots, factorization, and irreducibility in polynomial rings. Understanding this polynomial leads to deeper insights into how it behaves over various fields and its potential for having rational or irrational roots.
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