Below is a clear, simple, and exam-oriented explanation of Euclidean Domains (from Abstract Algebra / Discrete Structures). This will easily help you write long-answer questions in BCA/MCA/University exams.

⭐ Euclidean Domains

A Euclidean Domain (ED) is a special type of integral domain in which we can perform a division algorithm similar to integers.

In a Euclidean domain, we can divide one element by another and get:

[
a = bq + r, \quad \text{where } r = 0 \text{ or } \delta(r) < \delta(b)
]

Here δ is a special function (called the Euclidean function or valuation).

⭐ Formal Definition

An integral domain R is called a Euclidean domain if there is a function:

[
\delta : R \setminus {0} \to \mathbb{N}
]

(called a Euclidean function) such that for all (a, b \in R) with (b \neq 0):

There exist (q, r \in R) (quotient and remainder) such that:

[
a = bq + r
]

where either:

( r = 0 ), or
( \delta(r) < \delta(b) )

This condition guarantees we can perform Euclidean Algorithm to compute greatest common divisors (GCD).

⭐ Key Properties

Every Euclidean domain is an integral domain.
Every Euclidean domain is a principal ideal domain (PID).
Every Euclidean domain is a unique factorization domain (UFD).
It allows a division algorithm, just like integers.

⭐ Understanding Euclidean Function (δ)

The function δ assigns a non-negative integer to each non-zero element of the ring.

Examples:

For integers ℤ: δ(n) = |n|
For polynomials F[x]: δ(f) = degree of f

This function helps us compare remainder sizes.

⭐ Examples of Euclidean Domains

✔ 1. The ring of integers ℤ

Euclidean function:
[
\delta(n) = |n|
]

Division algorithm is standard:

[
a = bq + r,\quad 0 \le r < |b|
]

So ℤ is a Euclidean domain.

✔ 2. Polynomial rings over a field F[x]

Example: ℝ[x], ℚ[x], ℤ₅[x] etc.

Euclidean function:
[
\delta(f) = \deg(f)
]

Division algorithm:

[
f(x) = g(x)q(x) + r(x)
]

with deg(r) < deg(g).

✔ 3. Gaussian Integers (\mathbb{Z}[i])

Set:
[
\mathbb{Z}[i] = {a + bi : a, b \in \mathbb{Z}}
]

Euclidean function:
[
\delta(a+bi) = a^2 + b^2
]

So complex numbers of form a + bi (with integer components) form a Euclidean domain.

✔ 4. Euclidean Quadratic Fields

Examples:

(\mathbb{Z}[\sqrt{2}])
(\mathbb{Z}[\omega]) where (\omega = e^{2\pi i/3})

⭐ Non-Examples (Not Euclidean Domain)

✘ 1. ℤ[x] (polynomials with integer coefficients)

Not a Euclidean domain.

✘ 2. Some quadratic rings like (\mathbb{Z}[\sqrt{-5}])

Fails division algorithm.

⭐ Why Euclidean Domains are Important?

✔ They allow the Euclidean Algorithm

We can compute GCD like integers.

✔ Every ideal is principal

Thus, they are PIDs.

✔ Factorization is unique

Thus, they are UFDs (unique factorization domains).

✔ Good structure for solving equations

Diophantine equations, modular arithmetic, algebraic number theory.

⭐ Quick Exam Answer Summary

Definition:
A Euclidean domain is an integral domain (R) in which there is a function
(\delta : R \setminus {0} \to \mathbb{N})
such that for any (a, b \in R), (b \neq 0), there exist (q, r) satisfying:

[
a = bq + r,\quad \text{where } r = 0 \text{ or } \delta(r) < \delta(b)
]

Examples:

ℤ with δ(n) = |n|
F[x] with δ(f) = degree(f)
Gaussian integers ℤ[i]

Properties:

Every ED is a PID
Every ED is a UFD
GCD exists and can be computed

Euclidean domains