Polynomials

A field is a set which has an addition and a multiplication satisfying nine properties:
Let

1. Associativity of addition:
2. Commutativity of addition:
3. Additive identity: (See existential quantifier)
4. Additive inverse:
5. Associativity of multiplication:
6. Commutativity of multiplication:
7. Multiplicative identity:
8. Multiplicative inverse:
9. Distributivity:

Examples of fields are:

1. Rational numbers
2. Real numbers
3. Complex Numbers
4. The integers modulo a prime

Integers are NOT a field, because not every integer has an integer multiplicative inverse.

One of the most important facts about a field is, for all fields , and all , if then or

A polynomial in over the field is an expression of the form:

where .

• is a symbol called an indeterminate, and

Each individual is called a coefficient of the polynomial, and each individual expression of the form is called a term of the polynomial.

We use the notation to denote the set of all polynomials over field

Let be a polynomial in , where , and .

The polynomial is said to have degree and we write .
That is, the degree of a polynomial is the largest power of that has a non-zero coefficent.

The zero polynomial has all its coefficients equal to zero, and its degree is undefined

0. A constant polynomial that is either the zero polynomial or a polynomial of degree
1. Polynomials of degree 1 are called linear polynomials
2. Quadratic
3. Cubic
4. Quartic
5. Quintic
6. Sextic (or hexic)
7. Septic (or heptic)
8. Octic
9. Nonic
10. Decic

The polynomials and in are equal if and only if for all

A polynomial equation is an equation of the form

which is often written as where

An element is called a root of the polynomial if , meaning is a solution of the polynomial equation

The multiplicity of a root or a polynomial is the largest positive integer such that is a factor of

That is, the multiplicity of a root is the number of times it appears as a root.

For all fields , all polynomials , the remainder polynomial when is divided by is the constant polynomial

Find the remainder when is divided by (see Complex Numbers)

Instead of doing long division, we can use RT:

Hence the remainder is

For all fields , all polynomials , and all , the linear polynomial is a factor of the polynomial if and only if , meaning is a root of the polynomial

A polynomial is called monic if the non-zero coefficient of the highest power is

A polynomial in of positive degree is a reducible polynomial in when it can be written as the product of two polynomials in of positive degree.

Otherwise, we say that the polynomial is irreducible in

Given a polynomial , how do we know if it's reducible?

Let . If has no root in , is irreducible.

Consider . Factor this in

:
: Irreducible, since , and there are no real roots
: Since , there are also no roots in

For all polynomials , if , then is irreducible