Polynomial Rings

Polynomials

Let be a ring.

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.
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 an integral domain.

Proposition

Proof For any

Proposition

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

Evaluation Homomorphism

Suppose is a polynomial ring. Then the evaluation map given by for any is a ring homomorphism.

Proof This is easy to check.

The consequence of the above proposition is that if is a ring homomorphism, then there is a unique ring homomorphism that agrees with on constant polynomials and that maps to . More generally, given , there is a unique homomorphism that agrees with on constants and maps to .

Division with Remainder

Division Algorithm for Polynomials

Let be a commutative unital ring and with monic (i.e, the leading coefficient is a unit). 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.

A 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