Integral domain unit problem

[mathmajor2013]

Question: Show that if R is an integral domain then R^x=R[x]^x(or that the units in R[x] are precisely the units in R, but viewed as constant polynomials).

 

[Landau]

This is pretty straight-forward. What have you tried?

 

[Tom Gilroy]

Well, what happens when we multiply any non-constant polynomial in R[x] by any non-zero polynomial in R[x]?

 

[mathwonk]

could use concept of degree.

 

[blue_raver22]

The integral domain unit problem is a fundamental concept in algebra, and it relates to the units in a polynomial ring over an integral domain. In this case, we are interested in showing that the units in the polynomial ring R[x] are precisely the units in R, but viewed as constant polynomials.

To prove this, we first need to understand the definition of an integral domain. An integral domain is a commutative ring in which the product of any two non-zero elements is again non-zero. In other words, there are no zero divisors in an integral domain.

Now, let us consider an arbitrary element f(x) in R[x]. We can write f(x) as a polynomial of the form f(x) = a0 + a1x + a2x^2 + ... + anxn, where ai belongs to R and n is a non-negative integer. For f(x) to be a unit in R[x], there must exist a g(x) in R[x] such that f(x)g(x) = 1. This implies that the constant term a0 must be a unit in R. Otherwise, the product of a0 with any other term in the polynomial would result in a non-unit, contradicting the definition of a unit.

Now, let us consider the polynomial ring R[x]. Since R is an integral domain, all its units are precisely the elements a0 in R. Therefore, in R[x], the units are precisely the constant polynomials with the form a0 + 0x + 0x^2 + ... + 0xn. This shows that the units in R[x] are precisely the units in R, but viewed as constant polynomials.

In conclusion, we have shown that if R is an integral domain, then the units in R[x] are precisely the units in R, but viewed as constant polynomials. This result has important implications in algebra and can be applied in various mathematical problems and proofs.