Norms and Units of an Integral Domain

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SUMMARY

The discussion centers on the ring \(\mathbb{Z}[\sqrt{-5}]\) and its classification as a non-unique factorization domain (UFD). The norm is defined as \(N(a+\sqrt{-5}b)=a^2+5b^2\), and it is established that if \(z\) is a unit, then \(N(z)=1\). The participants clarify that reciprocal complex elements cannot exist in this ring due to the minimum value of the norm being 1, thus solidifying the understanding of units within this integral domain.

PREREQUISITES
  • Understanding of unique factorization domains (UFD)
  • Familiarity with algebraic structures, specifically integral domains
  • Knowledge of norms in algebraic number theory
  • Basic concepts of complex numbers and their properties
NEXT STEPS
  • Study the properties of unique factorization domains in detail
  • Explore the implications of norms in algebraic number theory
  • Investigate other examples of non-UFDs, such as \(\mathbb{Z}[\sqrt{-d}]\) for various \(d\)
  • Learn about the structure of units in different algebraic rings
USEFUL FOR

Mathematicians, algebra students, and educators interested in advanced algebraic concepts, particularly those focusing on integral domains and unique factorization properties.

alexfloo
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In an example in my algebra text, (from the section on unique factorization domains) it is describing the ring \mathbb{Z}[\sqrt{-5}], and demonstrating that it is not a UFD. It starts by giving the norm

N(a+\sqrt{-5}b)=a^2+5b^2.

It remarks that if zw=1, then N(z)N(w)=1, and then it goes on immediately to say that:

Therefore, if z=a+\sqrt{-5}b is a unit, then N(z)=a^2+5b^2=1.

This certainly seems plausible, but I don't see that he's actually proved it. It's not evident that we could not have any pair of reciprocal complex elements of \mathbb{Z}[\sqrt{-5}], which would then be units (which is what he appears to be assuming).
 
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Nevermind, I see now. We can't have reciprocal complex elements because the smallest nonzero value the norm can have is 1.
 

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