Norms and Units of an Integral Domain

In summary, the conversation discusses an example in an algebra text about unique factorization domains and specifically focuses on the ring \mathbb{Z}[\sqrt{-5}]. The text explains that this ring is not a UFD and gives the norm N(a+\sqrt{-5}b)=a^2+5b^2. It then mentions that if zw=1, then N(z)N(w)=1 and concludes that if z is a unit, then N(z)=1. The speaker initially questions the proof but later realizes that reciprocal complex elements are not possible due to the minimum value of 1 for the norm.
  • #1
alexfloo
192
0
In an example in my algebra text, (from the section on unique factorization domains) it is describing the ring [itex]\mathbb{Z}[\sqrt{-5}][/itex], and demonstrating that it is not a UFD. It starts by giving the norm

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

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

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

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 [itex]\mathbb{Z}[\sqrt{-5}][/itex], which would then be units (which is what he appears to be assuming).
 
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  • #2
Nevermind, I see now. We can't have reciprocal complex elements because the smallest nonzero value the norm can have is 1.
 

Related to Norms and Units of an Integral Domain

1. What is an integral domain?

An integral domain is a mathematical structure that consists of a set of elements and two binary operations, addition and multiplication. It is a commutative ring with unity, meaning that the operations of addition and multiplication follow the usual rules of commutativity and associativity, and there exists an identity element for each operation.

2. What are norms in an integral domain?

In an integral domain, a norm is a function that assigns a non-negative value to each element in the domain. It satisfies the properties of non-negativity, definiteness, and multiplicativity. Norms can be used to measure the size or magnitude of elements in the domain.

3. How are norms related to units in an integral domain?

Units in an integral domain are elements that have a multiplicative inverse, meaning they can be multiplied by another element to give the identity element. In some cases, the norm of a unit may be defined as 1, while for other elements it may be a non-zero value. Norms and units are closely related, as they both involve the concept of multiplication and the identity element.

4. What are the different types of units in an integral domain?

There are two types of units in an integral domain: invertible elements and zero divisors. Invertible elements are units that have a multiplicative inverse, while zero divisors are elements that multiply to give the zero element. Invertible elements are important in the study of norms and units, while zero divisors are not typically considered units.

5. How are units and norms used in real-world applications?

Units and norms are used in many different fields, such as physics, engineering, and economics. In physics, norms are used to measure physical quantities such as distance, time, and energy. In engineering, units are used to measure dimensions and quantities in design and construction. In economics, norms and units are used to measure and compare different currencies and monetary values. They are essential tools for solving real-world problems and making accurate calculations.

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