Question

Let \(F\) be a field and \(x, y\) indeterminates. Prove that the two ideals \(\langle x, y\rangle\) and \(\langle\operatorname{gcd}(x, y)\rangle\) in \(F[x, y]\) are distinct, and conclude that \(F[x, y]\) is not Euclidean.

Short answer

The ideals \(\langle x, y\rangle\) and \(\langle 1\rangle\) are distinct, proving \(F[x, y]\) is not Euclidean.

Step-by-step solution

Understanding the Problem

We are given two ideals \(\langle x, y\rangle\) and \(\langle\operatorname{gcd}(x, y)\rangle\) in the polynomial ring \(F[x, y]\). We need to prove that these ideals are distinct to conclude that \(F[x, y]\) is not Euclidean.

Exploring Ideal \(\langle x, y\rangle\)

The ideal \(\langle x, y\rangle\) is generated by the polynomials \(x\) and \(y\). This means every polynomial in this ideal can be expressed as \(ax + by\) where \(a, b \in F[x, y]\).

Exploring Ideal \(\langle\operatorname{gcd}(x, y)\rangle\)

Next, consider \(\operatorname{gcd}(x, y)\). Since \(x\) and \(y\) have no common factor other than 1 in returns of content in \(F[x, y]\), the gcd of these polynomials considered as elements of \(F[x, y]\) is 1. Thus, \(\langle\operatorname{gcd}(x, y)\rangle = \langle 1\rangle\), which is the entire ring \(F[x, y]\).

Comparing the Ideals

Since \(\langle 1\rangle\) is the entire ring, any polynomial in \(F[x, y]\) is in \(\langle\operatorname{gcd}(x, y)\rangle\). However, not every polynomial is necessarily in \(\langle x, y\rangle\), as polynomials that do not involve \(x\) or \(y\) cannot be expressed as \(ax + by\).

Conclusion on Distinct Ideals

Since \(\langle x, y\rangle\) contains just those polynomials expressible as \(ax + by\), and \(\langle 1\rangle\) is the whole ring, we conclude \(\langle x, y\rangle eq \langle\operatorname{gcd}(x, y)\rangle\). Thus, \(F[x, y]\) cannot be a Euclidean domain because in a Euclidean domain, every ideal would be principal and we have shown these two specific ideals are not equal.

Key concepts

Polynomial Ring

A polynomial ring is a type of ring formed by polynomials with coefficients from a given field or ring. When we talk about a polynomial ring like \(F[x, y]\), it means we're considering polynomials in two indeterminates, \(x\) and \(y\), with coefficients from a field \(F\). This allows us to perform addition and multiplication with these polynomials, much like with numbers.

• The elements of \(F[x, y]\) are polynomials in \(x\) and \(y\) with coefficients in \(F\).
• The operations of addition and multiplication follow the usual rules for polynomials, just extended to two variables.
• A polynomial ring can have polynomial equations representing geometric objects in algebraic geometry.

Breaking the polynomials into a ring structure helps to extend arithmetic techniques like division, GCD, and others from numbers to more complex expressions. Working in this multi-variable setting lays the groundwork for deeper studies in algebra and geometry.

Ideals in Algebra

Ideals are crucial concepts in algebra and specifically in ring theory. An ideal is a special subset of a ring that enables the construction of quotient rings and supports homomorphisms. Given a ring \(R\), an ideal \(I\) of \(R\) has these properties:

• For any elements \(a, b\) in \(I\), the sum \(a+b\) is in \(I\).
• For any element \(a\) in \(I\) and \(r\) in \(R\), the product \(ar\) is also in \(I\).

In our exercise, the ideals \(\langle x, y \rangle\) and \(\langle \operatorname{gcd}(x, y) \rangle\) are subsets of the polynomial ring \(F[x, y]\). These ideals are generated by their respective elements:

• \(\langle x, y \rangle\) contains all polynomials of the form \(ax + by\) where \(a, b \in F[x, y]\).
• \(\langle \operatorname{gcd}(x, y) \rangle = \langle 1 \rangle\) includes every polynomial in \(F[x, y]\) because 1 generates the whole ring.

Understanding ideals in algebra allows us to navigate toward the concept of factor rings and modular arithmetic, expanding our algebraic toolkit.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, extends from numbers to polynomials. Normally, the GCD of two integers is the largest integer dividing both numbers. In the context of polynomials, it is the highest degree monic polynomial which divides two polynomials completely:

• It provides a tool for simplifying fractions of polynomials.
• It helps analyze the structure of polynomial rings.

In our particular case with \(x\) and \(y\) in the polynomial ring \(F[x, y]\), because they are indeterminates, their only common divisor is 1. Hence, \(\operatorname{gcd}(x, y) = 1\). This makes the ideal generated by this GCD, \(\langle \operatorname{gcd}(x, y) \rangle\), the entire ring \(F[x, y]\). This is key in distinguishing the properties of the polynomial ring used in the exercise.

Fields in Algebra

Fields are foundational algebraic structures within which we perform arithmetic operations of addition, subtraction, multiplication, and division, except division by zero. A familiar example of a field is the set of rational numbers, \(\mathbb{Q}\), but fields extend beyond that.

• Fields are commutative, which means the order of numbers matters for operations.
• Fields have identity elements for both addition and multiplication.
• Each non-zero element in a field has a multiplicative inverse.

In this exercise, the base coefficients of the polynomials in \(F[x, y]\) come from a field \(F\). Working with fields ensures that we can carry out the division accurately when discussing ideals and polynomials. An understanding of fields allows clear insights into the nature of algebraic structures, extending to solving equations and exploring algebraic extensions.