Rings and Fields: Understanding Polynomials as a Commutative Ring with Unity

[oddiseas]

Homework Statement

is the set of all polynomials a ring,and a fieldd.Is is commutative and does it have unity

Homework Equations

The Attempt at a Solution

now if we add or multiply any polynomials we get a polynomial. So it is a ring, but i am not sure what the multiplicative inverse is or whether every non zero element has an inverse to constitute a field.

 

[HallsofIvy]

Multiplication is just ordinary multiplication of polynomials and the identity is the "constant" polynomial p(x)= 1 for all x. Does there exist a polynomial p(x) such that (x-1)(p(x))= 1? Think about the degree of p and what multiplication of polynomials does to degrees.

 

[oddiseas]

i am thinking that we do not have multiplicative inverses because there is no polynomial that would give one, we would need to use a rational function, is this a correct assesmnent?

 

[HallsofIvy]

Yes, but to give a complete answer, you need to say why "no polynomial would give one" (I presume you mean there is no polynomial, p(x), such that (x-1)p(x)= 1.)

Again, think of the "degree" of polynomials. The degree of x- 1 is 1. What could the degree of (x-1)p(x) be? Could it be equal to 0, the degree of the constant polynomial, 1?

 

[oddiseas]

ok thanks