# Euclidean Ring of Z[\zeta]: Unconventional Technique

• gonzo
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.
gonzo
Let
$$\displaystyle{\zeta = e^{{2\pi i} \over 5}}$$
I need to show that $Z[\zeta]$ is a Euclidean ring.

The only useful technique I know about is showing that given an element $\epsilon \in Q(\zeta)$ we can always find $\beta \in Z[\zeta]$ such that $N(\epsilon - \beta) < 1$ (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?

Last edited:
show it has a thingummmy - euclidean function, can't remember the precise name, that might help.

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.

## 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.

• Linear and Abstract Algebra
Replies
7
Views
838
• Classical Physics
Replies
4
Views
676
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
1
Views
854
• Linear and Abstract Algebra
Replies
11
Views
1K
• Linear and Abstract Algebra
Replies
9
Views
1K
• Other Physics Topics
Replies
2
Views
4K
• Linear and Abstract Algebra
Replies
11
Views
1K
• Linear and Abstract Algebra
Replies
2
Views
1K
• General Math
Replies
5
Views
3K