Arithmetic with Polynomials

Remark

is similar to

Addition and Multiplication of Polynomials

Definition

Let and

Then:

where for and .
That is, when adding polynomials, we add their coefficients.

where
for and where for , and for
That is, when multiplying polynomials, we have to multiply each term of one polynomial by each term of the other polynomial, INCLUDING the indeterminants (and their powers add up).

This is also known as expanding.

(See Summation and Product Notation)

Remark

This is a super technical definition which is practically useless. Adding and multiplying polynomials is very intuitive.

Example

In , for and :

In , for and

In , for and (see congruence class)

Degree of a Product

Lemma

For all fields , and all non-zero polynomials and in , we have:

Proof

Let be an arbitrary field, and and be arbitrary non-negative integers
Let and be arbitrary polynomials in of degree and respectively
So we have and
Now from the definition of multiplications, we have , where
Since and are non-zero elements of the field , it follows that is also non-zero
We therefore conclude that

Division Algorithm for Polynomials (DAP)

See division algorithm

Proposition

For all fields , and all polynomials and in and not the zero polynomial, there exists unique polynomials and in such that

where is the zero polynomial, or .

Divisibility in

Definition

For polynomials and over , we say that divides or is a factor of , if there exists a polynomial such that .
We write .

That is, divides when the remainder from DAP is the zero polynomial, or when and are both the zero polynomial.

See Polynomial Long Division

Unique Factorization Theorem on Polynomials (UFTP)

See unique factorization theorem

Theorem

For all polynomials can be expressed uniquely as a product of monic, irreducible polynomials and a non-zero constant.

Alternatively, all monic polynomials can be expressed uniquely as a product of monic irreducible polynomials without a non-zero constant.