Determining U(Z[x]) & U(R[x]) Rings

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In summary, the conversation revolves around determining the units in the rings U(Z[x]) and U(R[x]), which are related to abstract algebra. For U(Z[x]), it is mentioned that f(x)=1 and g(x)=-1 are potential units, but the reasoning behind this is unclear. For U(R[x]), there are some considerations about the degree of the polynomials and their units. The person is seeking help and guidance in solving these problems.
  • #1
joekoviously
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I am having a problem with some abstract algebra and I was wondering if anyone could help and give me some insight. The problem is as follows:

Give an explanation for your answer, long proof not needed:
Determine U(Z[x])
Determine U(R[x])

These are in regards to rings. I know for U(Z[x]) it is something like f(x)=1 g(x)=-1 but I don't know why.

As for U(R[x]) I am rather stuck. Any help or nudging in the right direction would be greatly appreciated.
 
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  • #2
For the first case suppose [tex]f \in \mathbb{Z}[x][/tex] is a unit. Then there exists another element [tex]g \in \mathbb{Z}[x][/tex] such that fg=1. If deg f > 0 what can you say about the degree of fg? Can fg be the multiplicative identity when deg fg >0? If deg f = 0 what can we say about f and g?

Similar considerations suffice for the ring of polynomials with real coefficients.
 

FAQ: Determining U(Z[x]) & U(R[x]) Rings

What is the difference between U(Z[x]) and U(R[x]) rings?

U(Z[x]) and U(R[x]) are both rings that contain elements of the form a + bx, where a and b are integers in the case of U(Z[x]), and a and b are real numbers in the case of U(R[x]). The main difference between the two is that U(Z[x]) is a ring of polynomials with integer coefficients, while U(R[x]) is a ring of polynomials with real coefficients.

How do you determine if a ring is a U(Z[x]) or U(R[x]) ring?

To determine if a ring is a U(Z[x]) or U(R[x]) ring, you need to check if all the elements in the ring can be written in the form a + bx, where a and b are integers for U(Z[x]) and real numbers for U(R[x]). If this is true, then the ring is a U(Z[x]) or U(R[x]) ring.

Can a U(Z[x]) or U(R[x]) ring have elements that are not in the form a + bx?

No, by definition, a U(Z[x]) or U(R[x]) ring consists only of elements in the form a + bx. If a ring has elements that cannot be written in this form, then it is not a U(Z[x]) or U(R[x]) ring.

How are the operations of addition and multiplication defined in U(Z[x]) and U(R[x]) rings?

In both U(Z[x]) and U(R[x]) rings, addition and multiplication are defined in the same way as they are in polynomial rings. Addition is simply the addition of the coefficients of like terms, and multiplication is the distribution and combination of all possible terms.

What are some examples of elements in U(Z[x]) and U(R[x]) rings?

Examples of elements in U(Z[x]) include 3 + 2x, -5 + 7x^3, and 0 + 4x^2. Examples of elements in U(R[x]) include 2.5 + 1.2x, -3.8 + 6.2x^2, and 0.5 + 0.3x^3. In both cases, the first element is a constant term and the second element is the coefficient of the variable x.

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