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] set of all polynomials with coefficients from a ring R
Coefficients come from a ring R (integers, real numbers, complex numbers)
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)
Polynomial rings have a unity element 1
Polynomial rings are integral domains if the coefficient ring R is an
No zero divisors: if f(x)⋅g(x)=0, then either f(x)=0 or g(x)=0
Example: Z[x] is an integral domain, but Z6[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+2x2−5 has degree 4
coefficient of the highest degree term
Example: In f(x)=3x4+2x2−5, the leading coefficient is 3
Zero polynomial has degree −∞ 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, but reducible over C as (x+i)(x−i)
Irreducibility depends on the field being considered
Irreducibility over specific fields
Over the field of real numbers R, a polynomial is irreducible if and only if it is:
Linear (degree 1)
Quadratic (degree 2) with a negative discriminant (b2−4ac<0)
Over the field of complex numbers C, every polynomial of degree greater than 0 is reducible
Fundamental Theorem of Algebra: every non-constant polynomial has a in C
sufficient condition for a polynomial to be irreducible over the field of rational numbers Q
If f(x)=anxn+an−1xn−1+⋯+a1x+a0 with integer coefficients, and there exists a prime p such that:
p divides a0,a1,…,an−1
p does not divide an
p2 does not divide a0
Then f(x) is irreducible over Q
Example: f(x)=x3+3x+6 is irreducible over Q by Eisenstein's criterion with p=3
The polynomial xp−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: (3x2+2x−1)+(2x2−3x+4)=5x2−x+3
Polynomial subtraction performed by subtracting the coefficients of like terms
Example: (3x2+2x−1)−(2x2−3x+4)=x2+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
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))
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))
Multiplication of polynomials is commutative, associative, and distributive over addition
Polynomial division and GCD
Division algorithm
algorithm for polynomials: given two polynomials f(x) and g(x) with g(x)=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)=x3+2x2−3x+1 by g(x)=x2+1 gives:
q(x)=x+1 and r(x)=x+2
f(x)=(x2+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+2 and g(x)=x2−1 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)=x3−3x+2 and g(x)=x2−1 with d(x)=x−1, we have: