Question

Let \(F\) be a field. Prove that the ideals \(I=\left\langle x+x y, y+x y, x^2, y^2\right\rangle\) and \(J=(x, y)\) in \(F[x, y]\) are identical. Your proof should also work if char \(F=2\). Hint: It is sufficient to prove that the generators of each ideal belong to the other ideal.

Short answer

The ideals \( I \) and \( J \) are identical because their generators are contained within each other.

Step-by-step solution

Understanding Ideals

Ideals are specific subsets of rings, and in this case, we are working within the polynomial ring \( F[x, y] \). The ideal \( J = (x, y) \) is generated by the elements \( x \) and \( y \), meaning any polynomial in \( J \) can be expressed as \( ax + by \) for some \( a, b \in F[x, y] \).

Generators of Ideal I

The ideal \( I = \langle x + xy, y + xy, x^2, y^2 \rangle \) is generated by the polynomials \( x + xy, y + xy, x^2, \) and \( y^2 \). To prove the ideals are identical, we need to show each generator of \( I \) can be expressed in terms of the generators of \( J \).

Show that Generators of I are in J

Since \( x + xy = x(1+y) \) and \( y + xy = y(1+x) \), if \( \text{char}(F) = 2 \), then \( 1+y = 1+x = 0 \), thus these terms cancel out to be zero, giving \( x+xy = x \) and \( y+xy = y \). This implies that \( x^2 = x \cdot x \) and \( y^2 = y \cdot y \) are also in \( J \). Therefore, all generators of \( I \) can be written as combinations of \( x \) and \( y \), hence they belong to \( J \).

Show that Generators of J are in I

We need to show \( x = (x + xy) - y \cdot x \) and \( y = (y + xy) - x \cdot y \) are in \( I \). Notice that \( x = 1 \cdot (x + xy) + 0 \cdot (y + xy) - x \cdot y \), which is composed of elements all in \( I \), likewise for \( y \). Therefore, \( x \) and \( y \) belong to \( I \).

Conclusion

Since the generators of \( I \) can be expressed in terms of the generators of \( J \) and vice versa, the ideals \( I \) and \( J \) are identical. This works even when \( \text{char}(F) = 2 \), adhering to the simplifications arising from \( 1+x = 0 \) or \( 1+y = 0 \).

Key concepts

Field Theory

Field theory is an exciting part of algebra that explores the fascinating world of fields. A field is a set equipped with two operations, addition and multiplication, where you can perform arithmetic like in the rational or real numbers. Every field must have a zero (identity for addition) and a one (identity for multiplication), and every non-zero element must have a multiplicative inverse. It's like working in a mathematical playground where division is always possible (except by zero).

Fields have additional properties that make them incredibly useful in algebra. For instance:

• The commutative property holds for both addition and multiplication.
• Associative properties are also present, ensuring consistency in operations.
• Distributive law ties addition and multiplication together nicely.
• Each element has an additive inverse, providing a perfect inverse world for subtraction.

Field theory becomes particularly engaging when combined with polynomial rings, offering us deeper insights into algebraic structures.

Polynomial Rings

Polynomial rings extend the concept of numbers by allowing the variables to take on 'ring-like' structures. In fields, polynomial rings become quite interesting because fields allow divisions, but polynomial rings don't allow division of variables. Imagine you have expressions like \(x + 3y\) or \(x^2 + xy + 7\); these belong to polynomial rings.

Here's what makes polynomial rings special:

• You can add, subtract, and multiply polynomials as you do with numbers, but division might not always work as expected.
• Polynomial rings over fields maintain several properties of fields, like having an additive identity (the zero polynomial) and a multiplicative identity (the number 1).
• Within a polynomial ring, any polynomial can be examined through its coefficients, and operations perform in a way that preserves the coefficients' structure.

Understanding polynomial rings is vital when dealing with concepts like ideals, which are subsets of rings that are closed under ring operations and exhibit specific properties.

Characteristic of a Field

The characteristic of a field is a fundamental property that tells us how a field behaves under addition. It's like finding out what happens after repeatedly adding the number one to itself.

Here's what you need to know about the characteristic:

• If you add one to itself repeatedly and eventually get zero, the number of times you added is the characteristic of the field. For example, if adding 1 three times gives zero (so \(1 + 1 + 1 = 0\)), the characteristic is 3.
• Fields with characteristic zero never hit zero this way, like the field of rational numbers, for instance, which behave like the usual counting numbers.
• The concept plays a critical role when considering modular arithmetic, especially in fields like finite fields where it influences structure dramatically.
• When dealing with polynomials and ideals in a field, the characteristic helps determine how elements like \(1 + x = 0\) simplify under the given field's rules.

The characteristic provides insights into the field's nature, influencing whether certain simplifications or equations hold, directly affecting polynomial operations and ideal properties.