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

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SUMMARY

The discussion focuses on determining the unit groups U(Z[x]) and U(R[x]) in the context of abstract algebra. It is established that U(Z[x]) consists of the constant polynomials f(x) = 1 and g(x) = -1, as these are the only units in the ring of polynomials with integer coefficients. For U(R[x]), the analysis indicates that any non-zero constant polynomial is a unit, as it can be multiplied by its reciprocal to yield the multiplicative identity. The discussion emphasizes the significance of polynomial degrees in identifying units within these rings.

PREREQUISITES
  • Understanding of abstract algebra concepts, specifically rings and units.
  • Familiarity with polynomial functions and their properties.
  • Knowledge of integer and real number coefficients in polynomial rings.
  • Basic grasp of polynomial degree and its implications in algebraic structures.
NEXT STEPS
  • Study the structure of U(Z[x]) in detail, focusing on the implications of polynomial degrees.
  • Explore the properties of U(R[x]) and the role of constant polynomials as units.
  • Investigate the concept of units in other polynomial rings, such as U(Q[x]) and U(F[x]) for fields F.
  • Review the fundamental theorem of algebra and its relation to polynomial units.
USEFUL FOR

This discussion is beneficial for students and researchers in abstract algebra, particularly those studying polynomial rings and their unit groups. It is also useful for educators seeking to clarify concepts related to units in algebraic structures.

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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For the first case suppose f \in \mathbb{Z}[x] is a unit. Then there exists another element g \in \mathbb{Z}[x] 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.
 

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