Are Units in R[x] Exactly the Same as Units in R When R is an Integral Domain?

In summary, the Integral Domain Unit Problem is a mathematical problem that asks whether a given integral domain has a multiplicative identity element, also known as a unit. It is important because it helps classify and understand different types of integral domains and has implications in other areas of mathematics. The difficulty of solving this problem varies depending on the integral domain being studied. Examples of integral domains with and without unit elements include integers, rational numbers, real numbers, complex numbers, and the ring of polynomials with coefficients in a finite field. This concept can also be extended to other algebraic structures, such as rings, fields, and modules.
  • #1
mathmajor2013
26
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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).
 
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  • #2
This is pretty straight-forward. What have you tried?
 
  • #3
Well, what happens when we multiply any non-constant polynomial in R[x] by any non-zero polynomial in R[x]?
 
  • #4
could use concept of degree.
 
  • #5


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.
 

FAQ: Are Units in R[x] Exactly the Same as Units in R When R is an Integral Domain?

What is the Integral Domain Unit Problem?

The Integral Domain Unit Problem is a mathematical problem that asks whether a given integral domain has a multiplicative identity element, also known as a unit. In other words, it asks if there is an element in the domain that when multiplied with any other element, produces the same element.

Why is the Integral Domain Unit Problem important?

The Integral Domain Unit Problem is important because it helps to classify and understand different types of integral domains. It also has important implications in other areas of mathematics, such as algebraic geometry and number theory.

Is solving the Integral Domain Unit Problem difficult?

The difficulty of solving the Integral Domain Unit Problem depends on the specific integral domain being studied. For some integral domains, the problem can be easily solved, while for others, it remains an open research question.

What are some examples of integral domains with and without unit elements?

Examples of integral domains with a unit element include the integers, rational numbers, real numbers, and complex numbers. An example of an integral domain without a unit element is the ring of polynomials with coefficients in a finite field.

Can the Integral Domain Unit Problem be extended to other algebraic structures?

Yes, the concept of a unit element can be extended to other algebraic structures, such as rings, fields, and modules. The corresponding problem would then be to determine if a given structure has a unit element.

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