Euclidean Ring of Z[\zeta]: Unconventional Technique

In summary, the conversation discusses proving that Z[\zeta] is a Euclidean ring using a standard norm for the euclidean function. The speaker mentions a technique involving finding an expression for the norm and choosing beta such that the difference of each basis element is less than 1/2. However, they are having trouble applying this method to the given expression of the norm. They mention a potential solution using a specific euclidean function, but the speaker is unsure of how to proceed with this approach.
  • #1
gonzo
277
0
Let
[tex]\displaystyle{\zeta = e^{{2\pi i} \over 5}}[/tex]
I need to show that [itex]Z[\zeta][/itex] is a Euclidean ring.

The only useful technique I know about is showing that given an element [itex]\epsilon \in Q(\zeta)[/itex] we can always find [itex]\beta \in Z[\zeta][/itex] such that [itex]N(\epsilon - \beta) < 1[/itex] (using the standard norm for the euclidean function).

This usually involves finding a general expression for the norm and then saying that you can choose beta such that the difference of each basis element is less than 1/2, and then showing that this means you can also get the norm less than 1.

However, the expression I got for the norm here didn't seem to lend itself to this method.

Any suggestions on how to do this?
 
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  • #2
show it has a thingummmy - euclidean function, can't remember the precise name, that might help.
 
  • #3
Of course, but that's the point. The problem is I can't show the Norm is less than one if the coefficients are less than 1/2 and don't know any other techniques.
 

Related to Euclidean Ring of Z[\zeta]: Unconventional Technique

1. What is the Euclidean Ring of Z[ζ]?

The Euclidean Ring of Z[ζ] is a mathematical structure that is used to study the properties of algebraic numbers. It is based on the concept of a Euclidean ring, which is a type of algebraic structure that allows for division with remainders.

2. How is the Euclidean Ring of Z[ζ] different from other rings?

The Euclidean Ring of Z[ζ] is different from other rings because it is based on the algebraic numbers, which are complex numbers that are solutions to polynomial equations with integer coefficients. This makes it a more specialized and powerful tool for studying algebraic structures.

3. What is the unconventional technique used in the Euclidean Ring of Z[ζ]?

The unconventional technique in the Euclidean Ring of Z[ζ] is the use of algebraic numbers, specifically the use of the complex number ζ as the generator of the ring. This allows for the efficient and elegant study of algebraic structures and their properties.

4. What are some practical applications of the Euclidean Ring of Z[ζ]?

The Euclidean Ring of Z[ζ] has applications in various fields such as number theory, cryptography, and coding theory. It is also used in the study of elliptic curves and algebraic geometry.

5. Is the Euclidean Ring of Z[ζ] a commonly used concept in mathematics?

The Euclidean Ring of Z[ζ] is a more specialized and advanced concept in mathematics, so it may not be as commonly used as other mathematical structures. However, it is a powerful and important tool in the study of algebraic structures and has applications in various fields of mathematics.

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