Finding the units (number theory)

[Firepanda]

Homework Statement

Prove that the ring R of polynomials with real coefficients (i.e. f(x) = a₀ + a₁x + ... + a_nxⁿ, a_i real, are elements of R) has only the constant term a₀ as the group of units, providing the constant term isn't zero.

Homework Equations

u is a unit if there exists a v such that uv=1

The Attempt at a Solution

Clearly all the constant terms are units of the ring as we are dealing with the real numbers.

Claim: There are more than just these, and that a polynomial f(x) = a₀ + a₁x + ... + a_nxⁿ is invertible.

=> there must exist a g(x) = b₀ + b₁x + ... + b_nxⁿ

such that f(x).g(x) = 1

f(x).g(x) = b₀(a₀ + a₁x + ...) + b₁x(a₀ + a₁x + ...) + b₂x²(a₀ + a₁x + ...) + ... = 1

Not too sure where to go from here.. Sorry if my notation is a little off but hopefully you can understand

Thanks

 

[JSuarez]

R[x] is the ring of polynomials over a field: R. In this case, what is the degree of the product of two polynomials? What must be the degree of a unit?

 

[Dick]

If f(x) and g(x) aren't constants, then each one has a highest degree term with a coefficient that's nonzero, right? What's the coefficient of the highest degree term of f(x)*g(x)?

 

[Firepanda]

R[x] is the ring of polynomials over a field: R. In this case, what is the degree of the product of two polynomials? What must be the degree of a unit?

If f has degree n and g has degree m then the degree of the product is n+m which must be greater than or equal to 2, but the the degree of 1 is 0.

Surely some polynomial can equal 1 though can't it?

If f(x) and g(x) aren't constants, then each one has a highest degree term with a coefficient that's nonzero, right? What's the coefficient of the highest degree term of f(x)*g(x)?

coefficient a_nb_m, not too sure how this helps though

 

[Dick]

If an*bm is not zero, then f(x)*g(x) contains a term of the form an*bm*x^(n+m) and that's the only term of that degree. So the product is probably not 1.

 

[Firepanda]

But if these are real coefficients then maybe they're small enough so we can choose an x such that the product is equal to 1?

 

[Dick]

"1" has coefficient 0 for all powers of x greater than 0. Not "close to zero". Zero. I'm having a hard time picturing what you are thinking about here. The reals don't have any zero divisors.

 

[Dick]

But if these are real coefficients then maybe they're small enough so we can choose an x such that the product is equal to 1?

"x" isn't a number. It's a symbol. This question isn't about roots of a polynomial. Is that what you are thinking?

 

[Firepanda]

"x" isn't a number. It's a symbol. This question isn't about roots of a polynomial. Is that what you are thinking?

ye i was thinking that, guess that's why I'm having a hard time understanding

I can just go by the degrees of 1 and the product of the polynomials are different then as above?

Thanks

 

[Dick]

Yes, you can. Two polynomials in the "ring of polynomials" are equal only if all of their coefficients are equal. Look back at the definition of "ring of polynomials". Nothing to do with whether they are equal for some particular value of x.