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Euclidean Domain | Algebra - Mathematics PDF Download

A Euclidean domain is an integral domain R which can be equipped with a function

d : R\{0} → N

such that for all a ∈ R and b ≠ 0, b ∈ R we can write

a = qb + r

for some q, r ∈ R with r = 0 or d(r) < d(b).

For example. Z with d(n) = |n| is a Euclidean Domain; also, for any field k, k[X ] with d(f ) = deg(f ) is a Euclidean Domain. (WARNING: In the second example above, it is essential that k be a field.)

We shall prove that every Euclidean Domain is a Principal Ideal Domain (and so also a Unique Factorization Domain). This shows that for any field k, k[X ] has unique factorization into irreducibles. As a further example, we prove that Euclidean Domain | Algebra - Mathematics is a Euclidean Domain.

Proposition 1. In a Euclidean domain, every ideal is principal. 

Proof. Suppose R is a Euclidean domain and Euclidean Domain | Algebra - Mathematics . Then EITHER I = {0} = (0) OR we can take a ≠ 0 in I with d(a) least; then for any b ∈ I , we can write b = qa + r with r = 0 or d(r) < d(a); but r = q − ba ∈ I and so by minimality of d(a), r = 0; thus a|b and I = (a).

Corollary 2. If k is a field then every ideal in k[X ] is principal.

Corollary 3. Let k be a field. Then every polynomial in k[X ] can be factorized into primes=irreducibles, and the factorization is essential ly unique.

Corollary 4. Every element of the ring Euclidean Domain | Algebra - Mathematics can be factorized into primes= irreducibles, and the factorization is essential ly unique.

Proof. By Theorem 1, it is enough to show that Euclidean Domain | Algebra - Mathematics is a Euclidean Domain.To this end, define N : Euclidean Domain | Algebra - Mathematics → N by

Euclidean Domain | Algebra - Mathematics

Note that we can extend N to a function N : Q Euclidean Domain | Algebra - Mathematics → Q defined similarly by

Euclidean Domain | Algebra - Mathematics

Note also that given any Euclidean Domain | Algebra - Mathematics we have

Euclidean Domain | Algebra - Mathematics

Now, suppose we are given  Euclidean Domain | Algebra - Mathematics

Then

Euclidean Domain | Algebra - Mathematics

Pick  Euclidean Domain | Algebra - Mathematics

Euclidean Domain | Algebra - Mathematics

Euclidean Domain | Algebra - Mathematics

Euclidean Domain | Algebra - Mathematics

The document Euclidean Domain | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Euclidean Domain - Algebra - Mathematics

1. What is an Euclidean domain in mathematics?
Ans. An Euclidean domain is a type of mathematical structure in abstract algebra. It is a commutative ring equipped with a function called the Euclidean function or Euclidean norm. This function allows division with remainder, similar to the division algorithm in ordinary arithmetic.
2. How does the Euclidean algorithm relate to Euclidean domains?
Ans. The Euclidean algorithm is a fundamental algorithm used to find the greatest common divisor (GCD) of two numbers. In the context of Euclidean domains, the Euclidean algorithm is used to compute the GCD of elements in the domain. It takes advantage of the Euclidean norm to perform repeated divisions with remainder until the remainder becomes zero, at which point the last non-zero remainder is the GCD.
3. What are some examples of Euclidean domains?
Ans. Examples of Euclidean domains include the set of integers Z, the set of Gaussian integers Z[i] (complex numbers with integer coefficients), and the set of polynomials with integer coefficients Z[x]. In each of these cases, the Euclidean norm is defined in a way that allows for division with remainder.
4. How can Euclidean domains be used in solving equations or factorizing numbers?
Ans. Euclidean domains have useful properties that make them valuable in solving equations or factorizing numbers. For example, in a Euclidean domain, the Euclidean algorithm can be used to solve linear Diophantine equations or compute modular inverses. Additionally, the Euclidean algorithm can be extended to factorize numbers by repeatedly applying it to find the GCD of the number with different values.
5. Are all integral domains Euclidean domains?
Ans. No, not all integral domains are Euclidean domains. In fact, Euclidean domains are a more specific type of integral domain. An integral domain is a commutative ring where the product of any two non-zero elements is non-zero. While all Euclidean domains are integral domains, the converse is not true. For example, the ring of polynomials with real coefficients is an integral domain but not a Euclidean domain.
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