Closure Of A Field: Proving It's Still A Field

Hey guys! Let's dive into a cool problem from Resnick's book that touches on both Real Analysis and Field Theory. The heart of the matter revolves around showing that the closure of a field retains the properties that make it a field. This is one of those cool theoretical problems that really helps you solidify your understanding of what fields are all about and how topological concepts like closure interact with algebraic structures.

Understanding the Question

So, the core question is: If we have a field ${ F }$, and we take its closure (let's call it ${ \overline{F} }$), how do we show that ${ \overline{F} }$ is also a field? You might wonder, like many others, how the closure could be smaller than the original field. That's a valid point that often leads to confusion! The closure of a set, in any topological space, is the smallest closed set containing that set. In simpler terms, it includes all the limit points of the set. For a field, this means we're adding in all the values that sequences in the field converge to, which might not have been in the original field.

What Makes a Field a Field?

Before we get knee-deep in the proof, let's quickly recap what properties a set needs to satisfy to be considered a field. A field, in algebraic terms, is a set on which addition and multiplication are defined and satisfy a bunch of axioms. Here’s a quick rundown:

1. Closure under Addition and Multiplication: For any elements ${ a }$ and ${ b }$ in the field, ${ a + b }$ and ${ a \cdot b }$ must also be in the field.
2. Associativity of Addition and Multiplication: For any elements ${ a }$, ${ b }$, and ${ c }$ in the field, ${ (a + b) + c = a + (b + c) }$ and ${ (a \cdot b) \cdot c = a \cdot (b \cdot c) }$.
3. Commutativity of Addition and Multiplication: For any elements ${ a }$ and ${ b }$ in the field, ${ a + b = b + a }$ and ${ a \cdot b = b \cdot a }$.
4. Existence of Additive and Multiplicative Identities: There exist elements ${ 0 }$ and ${ 1 }$ in the field such that for any element ${ a }$ in the field, ${ a + 0 = a }$ and ${ a \cdot 1 = a }$.
5. Existence of Additive and Multiplicative Inverses: For every element ${ a }$ in the field, there exists an element ${ -a }$ such that ${ a + (-a) = 0 }$, and for every element ${ a }$ except ${ 0 }$, there exists an element ${ a^{-1} }$ such that ${ a \cdot a^{-1} = 1 }$.
6. Distributivity of Multiplication over Addition: For any elements ${ a }$, ${ b }$, and ${ c }$ in the field, ${ a \cdot (b + c) = a \cdot b + a \cdot c }$.

Proving the Closure is a Field

Now, let's show that the closure ${ \overline{F} }$ of a field ${ F }$ is also a field. This means we need to show that all the field axioms hold true for ${ \overline{F} }$.

Closure Under Addition and Multiplication

This is the crucial part. We need to show that if ${ x, y \in \overline{F} }$, then ${ x + y }$ and ${ x \cdot y }$ are also in ${ \overline{F} }$. Remember, if ${ x \in \overline{F} }$, it means that there exists a sequence ${ (x_n) }$ in ${ F }$ such that ${ x_n \to x }$ as ${ n \to \infty }$. Similarly, if ${ y \in \overline{F} }$, there exists a sequence ${ (y_n) }$ in ${ F }$ such that ${ y_n \to y }$ as ${ n \to \infty }$.

Since ${ F }$ is a field, ${ x_n + y_n }$ and ${ x_n \cdot y_n }$ are in ${ F }$ for all ${ n }$. Because the limit of the sum is the sum of the limits, we have:

${\lim_{n \to \infty} (x_n + y_n) = \lim_{n \to \infty} x_n + \lim_{n \to \infty} y_n = x + y}$

Similarly, the limit of the product is the product of the limits:

${\lim_{n \to \infty} (x_n \cdot y_n) = \lim_{n \to \infty} x_n \cdot \lim_{n \to \infty} y_n = x \cdot y}$

This shows that ${ x + y }$ and ${ x \cdot y }$ are limits of sequences in ${ F }$, and thus they belong to ${ \overline{F} }$. So, ${ \overline{F} }$ is closed under addition and multiplication.

Associativity and Commutativity

These properties hold true in ${ \overline{F} }$ because they hold true in ${ F }$, and the operations in ${ \overline{F} }$ are defined as limits of operations in ${ F }$. For example, for associativity of addition, let ${ x, y, z \in \overline{F} }$. Then there exist sequences ${ (x_n), (y_n), (z_n) }$ in ${ F }$ that converge to ${ x, y, z }$ respectively. Since ${ (x_n + y_n) + z_n = x_n + (y_n + z_n) }$ for all ${ n }$, it follows that:

${\lim_{n \to \infty} ((x_n + y_n) + z_n) = \lim_{n \to \infty} (x_n + (y_n + z_n))}$

Which implies:

${(x + y) + z = x + (y + z)}$

The same logic applies to commutativity and associativity of multiplication.

Existence of Additive and Multiplicative Identities

The additive identity ${ 0 }$ and the multiplicative identity ${ 1 }$ are already in ${ F }$ (since ${ F }$ is a field), and therefore, they are also in ${ \overline{F} }$. For any ${ x \in \overline{F} }$, ${ x + 0 = x }$ and ${ x \cdot 1 = x }$.

Existence of Additive and Multiplicative Inverses

For any ${ x \in \overline{F} }$, we need to show that ${ -x }$ is also in ${ \overline{F} }$. If ${ x \in \overline{F} }$, there exists a sequence ${ (x_n) }$ in ${ F }$ such that ${ x_n \to x }$. Since ${ F }$ is a field, ${ -x_n }$ is in ${ F }$ for all ${ n }$. Thus, the sequence ${ (-x_n) }$ is in ${ F }$, and ${ \lim_{n \to \infty} (-x_n) = -x }$. This means that ${ -x \in \overline{F} }$.

Similarly, for any ${ x \in \overline{F} }$ with ${ x \neq 0 }$, we need to show that ${ x^{-1} }$ is in ${ \overline{F} }$. Since ${ x \neq 0 }$, we can find a sequence ${ (x_n) }$ in ${ F }$ such that ${ x_n \to x }$ and ${ x_n \neq 0 }$ for sufficiently large ${ n }$. Then ${ x_n^{-1} }$ is in ${ F }$ for all such ${ n }$, and ${ \lim_{n \to \infty} x_n^{-1} = x^{-1} }$. Thus, ${ x^{-1} \in \overline{F} }$.

Distributivity

Distributivity of multiplication over addition holds in ${ \overline{F} }$ because it holds in ${ F }$. If ${ x, y, z \in \overline{F} }$, then there exist sequences ${ (x_n), (y_n), (z_n) }$ in ${ F }$ that converge to ${ x, y, z }$ respectively. Since ${ x_n \cdot (y_n + z_n) = x_n \cdot y_n + x_n \cdot z_n }$ for all ${ n }$, it follows that:

${\lim_{n \to \infty} (x_n \cdot (y_n + z_n)) = \lim_{n \to \infty} (x_n \cdot y_n + x_n \cdot z_n)}$

Which implies:

${x \cdot (y + z) = x \cdot y + x \cdot z}$

Conclusion

By showing that all the field axioms hold for ${ \overline{F} }$, we've proven that the closure of a field is indeed a field. This exercise is super helpful in understanding how algebraic structures behave under topological operations. Keep up the great work, and happy problem-solving!

Resnick, Chap 1 Prob 44

The original problem from Resnick's book, Chapter 1, Problem 44, likely aims to reinforce these concepts. The confusion about the closure being "smaller" than the original field usually stems from a misunderstanding of what closure entails—it's about including limit points, not removing elements. Therefore, the closure will always contain the original field and can potentially be larger if the field isn't already closed. Understanding this subtle nuance is key to mastering these types of problems. So, keep practicing and refining your understanding!