Definition of a Euclidean Domain ....

In summary, the definition of a Euclidean Domain varies between the books "The Basics of Abstract Algebra" by Bland and "Abstract Algebra" by Dummit and Foote. Bland defines it using two conditions, while Dummit and Foote only use one of Bland's conditions. This difference in definition may result in some structures being recognized as Euclidean Domains under Dummit and Foote's definition, but not under Bland's definition. However, both definitions still result in a Euclidean Domain with a norm function, which is an important property in the study of ideals and algebraic-number theory.
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In the book "The Basics of Abstract Algebra" Bland defines a Euclidean Domain using two conditions as follows:View attachment 8256
View attachment 8257In the book "Abstract Algebra"by Dummit and Foote we find that a Euclidean Domain is defined using only one of Bland's conditions ... as follows:View attachment 8258What are the consequences of these different definitions ... for example does D&F's definition allow some structures to be Euclidean Domains that are not recognized as such under Bland's definition ...Peter
 
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Let $N$ be the norm of the integral domain $R$ in F&D’s definition. Define another norm $N^\ast$ by
$$N^\ast(a)\ =\ \min_{b\in R\setminus\left\{0_R\right\}}N(ab).$$
Then $R$ is still a Euclidean domain with norm $N^\ast$ and $N^\ast$ is also a Euclidean valuation in Bland’s definition.

The most important property of a Euclidean domain is the second one in Bland; many results about Euclidean domains involve only the second property but not the first. IMHO the importance of the first property only shows up in the study of ideals and algebraic-number theory.
 

1. What is a Euclidean Domain?

A Euclidean Domain is a mathematical structure that consists of a set of elements and two operations, addition and multiplication, that follow the same rules as regular arithmetic. It is a type of ring that allows for division with remainder, making it useful for solving problems involving integers and polynomials.

2. What are the properties of a Euclidean Domain?

A Euclidean Domain must have the following properties: a well-defined addition and multiplication operation, closure under addition and multiplication, commutativity and associativity of addition and multiplication, existence of additive and multiplicative identities, and the existence of a Euclidean function that measures the "size" of elements in the domain.

3. How is a Euclidean Domain different from other types of rings?

A Euclidean Domain is unique because it allows for division with remainder, which is not possible in other types of rings such as integral domains or fields. This makes it useful for solving problems involving integers and polynomials, and it also has applications in number theory and algebraic geometry.

4. What is the Euclidean Algorithm and how is it related to Euclidean Domains?

The Euclidean Algorithm is a method for finding the greatest common divisor (GCD) of two elements in a Euclidean Domain. It utilizes the Euclidean function to repeatedly divide the larger element by the smaller element and then using the remainder as the new divisor until a remainder of 0 is reached. The final non-zero remainder is the GCD. This algorithm is only possible in Euclidean Domains where division with remainder is allowed.

5. What are some examples of Euclidean Domains?

Some common examples of Euclidean Domains include the ring of integers, the ring of polynomials with coefficients in a field, and the ring of Gaussian integers (complex numbers of the form a + bi where a and b are integers). Other examples include the ring of algebraic integers, the ring of quadratic integers, and the ring of Eisenstein integers.

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