Problem 18 - A First Course in Abstract Algebra

Question

Show that every field is a Eaclidean domain.

Step-by-Step Solution

Answer

Every field is a Euclidean domain because division is always possible with remainder zero.

Step 1: Definitions

To determine whether every field is a Euclidean domain, we start by defining both fields and Euclidean domains. A **field** is a set equipped with two operations, addition and multiplication, such that every nonzero element has an inverse under multiplication. A **Euclidean domain** is an integral domain with a Euclidean function \( \phi : D \setminus \{0\} \to \mathbb{N} \) that allows for a division algorithm.

Step 2: Properties of Fields

One important property of fields is that every nonzero element in a field has a multiplicative inverse. This means you can divide by any nonzero element in a field. Additionally, in a field, the only divisors of zero are zero itself.

Step 3: Euclidean Function in a Field

To show a field is a Euclidean domain, we need to specify a Euclidean function. We can choose \( \phi(a) = 1 \) for any nonzero element \( a \) in the field. Since fields allow division by any nonzero element, for any two elements \( a \) and \( b \) (with \( b = a \cdot q \) and remainder \( r = 0 \)), the classical division property holds as \( a = bq + r \) with \( r = 0 \).

Step 4: Verifying Euclidean Domain Conditions

In a Euclidean domain, for any two elements \( a \) and \( b \) (with \( a eq 0 \)), you need to be able to write \( b = a \cdot q + r \) where \( r \) is either zero or has a smaller Euclidean function value than \( a \). In a field, we can directly choose \( r = 0 \) since division is always possible, satisfying this requirement with \( b = a \cdot (b/a) \).

Step 5: Conclusion

Since we can define a Euclidean function \( \phi(a) = 1 \) for any nonzero element in a field and meet the condition \( b = a \cdot q + r \) with \( r = 0 \), all fields satisfy the criteria for being Euclidean domains.

Key Concepts

Euclidean domain

A Euclidean domain is a fascinating concept in the realm of mathematics. It is a kind of ring where you can apply a version of division, similar to how you divide integers. For a ring to be a Euclidean domain, it must be an integral domain, and there must exist a Euclidean function \( \phi \). This function maps nonzero elements to non-negative integers and allows you to perform division with a remainder.

• The division must be such that for any two elements \( a \) and \( b \) in the domain (where \( b e 0 \)), you can write \( a = bq + r \).
• Here, \( q \) is the quotient and \( r \) is the remainder.
• Importantly, \( r \) should be either zero or smaller in \( \phi \) value than \( b \).

Euclidean domains are crucial in simplifying calculations and proving further algebraic theorems. They are vital in number theory and algebra applications.

Integral domain

An integral domain is a fundamental algebraic structure with properties that resemble those of the integers. It's a commutative ring with a multiplicative identity element, usually denoted as \( 1 \), that does not have any zero divisors. What does that mean?

• Being commutative means that multiplication can occur in any order—so \( a \times b = b \times a \).
• A multiplicative identity ensures that there is a number (1 in this case) that doesn’t change other numbers when multiplied by them.
• No zero divisors mean that if \( ab = 0 \), either \( a \) or \( b \) must be zero.

Integral domains are a stepping stone towards understanding fields. Every field is an integral domain, but not every integral domain is a field.

Division algorithm

The division algorithm is a process or rule that is fundamental in Euclidean domains. It allows for a division of any two elements where the result includes a quotient and a remainder. How does it work?

• Given any elements \( a \) and \( b \) (where \( b e 0 \)) in a Euclidean domain, you can always find a quotient \( q \) and a remainder \( r \).
• The remainder \( r \) must be such that either \( r = 0 \) or it's smaller in value than \( b \) according to the Euclidean function \( \phi \).
• This procedure is similar to regular number division where \( a = bq + r \).

This algorithm lays the groundwork for advanced computations in algebra, like finding greatest common divisors or simplifying complex problems.

Multiplicative inverse

The concept of a multiplicative inverse plays a critical role in fields within mathematics. A multiplicative inverse of a number \( a \) is another number \( b \) such that when you multiply \( a \) by \( b \), the result is 1.

• In fields, every nonzero element must have a multiplicative inverse.
• This property implies that division is always possible within a field (except by zero, naturally).
• For instance, in the field of real numbers, the multiplicative inverse of 2 is \( \frac{1}{2} \) because \( 2 \times \frac{1}{2} = 1 \).

Understanding multiplicative inverses is essential as they assure the capability to perform division in fields, leading to ease in solving equations involving multiplication.