Polynomial Rings

Polynomials

Polynomial Ring

Let be a ring. The polynomial ring in one variable over is the set of sequences:Equivalently,with addition and multiplication .

Proposition

is a ring. The identity is the polynomial .

Degree and Leading Term

The degree of a polynomial , denoted , is the largest with . The element zero does not have a degree.

Leading Term & Monic

The term for is called the leading term of . The polynomial is called monic if the coefficient of the leading term is .

Proposition

For all , with equality if and only if is a integral domain.

Proposition

Proposition

If is an integral domain, then is an integral domain.

Division with Remainder

Theorem

Let be a ring and with monic. Then there are uniquely determined polynomials , such that and such that or . The polynomial is called the quotient and the polynomial is called the remainder.

Def Substitution Principle I and Polynomial Function Let be a ring and . The substitution of in is the elementA function is called a polynomial function if there exists such that for every .

Corollary

Let and . The remainder of division of by is . It follows that divides if and only if .

Proof We have unique such that and or . Hence divides if and only if .

Corollary A polynomial in of degree over a field (more generally, a domain) has at most roots.

Multivariable Polynomials

For and we write . The degree of is . The collection of all these is called the ring of polynomials in variables with coefficients in and denoted .

Proposition

Prop Every ideal in the polynomial ring over a field is principal. A non-zero ideal in is generated by the unique monic polynomial of lowest degree that it contains.

Proof

Def Greatest Common Divisor Let be a field and be non-zero elements. The greatest common divisor of and is the unique monic polynomial that generates the ideal . It has the properties:

3. If then
4. There are polynomials such that