Polynomial arithmetic


Polynomial arithmetic is a branch of algebra dealing with some properties of polynomials which share strong analogies with properties of number theory relative to integers.
It includes basic mathematical operations such as addition, subtraction, and multiplication, as well as more elaborate operations like Euclidean division, and properties related to roots of polynomials. The latter are essentially connected to the fact that the set K of univariate polynomials with coefficients in a field K is a commutative ring, such as the ring of integers.

Elementary operations on polynomials

Addition and subtraction of two polynomials are performed by adding or subtracting corresponding coefficients. If
then addition is defined as
Multiplication is performed much the same way as addition and subtraction, but instead by multiplying the corresponding coefficients. If then multiplication is defined as where. Note that we treat as zero for and that the degree of the product is equal to the sum of the degrees of the two polynomials.

Advanced polynomial arithmetics and comparison with number theory

Many fascinating properties of polynomials can be found when, thanks to the basic operations that can be performed on two polynomials and the underlying commutative ring structure of the set they live in, one tries to apply reasonings similar to those known from number theory.
To see this, one first needs to introduce two concepts: the notion of root of a polynomial and that of divisibility for pairs of polynomials.
If one considers a polynomial of a single variable X in a field K, and with coefficients in that field, a root of is an element of K such that
The second concept, divisibility of polynomials, allows to see a first analogy with number theory: a polynomial is said to divide another polynomial when the latter can be written as
with C being ALSO a polynomial. This definition is similar to divisibility for integers, and the fact that divides is also denoted.
The relation between both concepts above arises when noticing the following property: is a root of if and only if. Whereas one logical inclusion is obvious, the other relies on a more elaborate concept, the Euclidean division of polynomials, here again strongly reminding of the Euclidean division of integers.
From this it follows that one can define prime polynomials, as polynomials that cannot be divided by any other polynomials but 1 and themselves - here again the analogously with prime integers is manifest, and allows that some of the main definitions and theorems related to prime numbers and number theory have their counterpart in polynomial algebra. The most important result is the fundamental theorem of algebra, allowing for factorization of any polynomial as a product of prime ones. Worth mentioning is also the Bézout's identity in the context of polynomials. It states that two given polynomials P and Q have as greatest common divisor a third polynomial D, if and only if there exists polynomials U and V such that