The Real Closure of a Field

Oct 13, 2025 by miodeya.com 81 views

Introduction

In the realm of abstract algebra, particularly in the study of real fields and their properties, a fundamental problem has been the subject of intense research. This problem, known as Hilbert's 17th problem, deals with the representation of nonnegative polynomials in a real field as sums of squares of polynomials. In this article, we will delve into the solution to this problem, known as Artin's solution, and explore the implications of this result.

Background and History

Hilbert's 17th problem was first proposed by David Hilbert in 1900 as part of a list of 23 problems that he presented to the International Congress of Mathematicians in Paris. The problem was stated as follows: "Given a polynomial $f(x_1, \ldots, x_n)$ with real coefficients, is it possible to find a representation of $f$ as a sum of squares of rational functions?" This problem was significant not only because of its intrinsic interest but also because it had far-reaching implications for the study of real fields and their properties.

Artin's Solution

The solution to Hilbert's 17th problem was provided by Emil Artin in 1927. Artin's solution was a major breakthrough in the field of real fields and had significant implications for the study of nonnegative polynomials. In his solution, Artin showed that if $E$ is a real field admitting a unique ordering, then any polynomial $f \in E[X_1, \ldots, X_n]$ that is nonnegative at every point of the real closure $R$ of $E$ can be represented as a sum of squares of polynomials in $E[X_1, \ldots, X_n]$ .

The Real Closure of a Field

Before we can discuss Artin's solution in more detail, we need to understand the concept of the real closure of a field. The real closure of a field $E$ is a real field $R$ that contains $E$ as a subfield and is such that every nonempty set of elements of $R$ that is bounded above has a least upper bound in $R$ . In other words, the real closure of a field is a real field that is complete with respect to the ordering on the field.

The Proof of Artin's Solution

The proof of Artin's solution is based on a series of lemmas and theorems that establish the existence of a representation of nonnegative polynomials as sums of squares of polynomials. The key idea behind the proof is to show that if a polynomial $f$ is nonnegative at every point of the real closure $R$ of $E$ , then there exists a polynomial $g$ such that $f = g^2$ . This is done by using a series of algebraic manipulations and the properties of the real closure of a field.

The Representation Theorem

The representation theorem, which is the main result of Artin's solution, states that if $E$ is a real field admitting a unique ordering, then any polynomial $f \in E[X_1, \ldots, X_n]$ that is nonnegative at every point of the real closure $R$ of $E$ can be represented as a sum of squares of polynomials in $E[X_1, \ldots, X_n]$ . This theorem has far-reaching implications for the study of nonnegative polynomials and has been used in a wide range of applications, including algebraic geometry and optimization theory.

Implications of Artin's Solution

Artin's solution to Hilbert's 17th problem has had a significant impact on the study of real fields and their properties. The representation theorem, which is the main result of Artin's solution, has been used in a wide range of applications, including algebraic geometry and optimization theory. In addition, Artin's solution has led to a deeper understanding of the properties of nonnegative polynomials and has provided new insights into the study of real fields.

Conclusion

In conclusion, Artin's solution to Hilbert's 17th problem is a major breakthrough in the field of real fields and has had a significant impact on the study of nonnegative polynomials. The representation theorem, which is the main result of Artin's solution, has been used in a wide range of applications and has provided new insights into the study of real fields. As we continue to explore the properties of real fields and their applications, Artin's solution will remain an important and influential result in the field.

References

Artin, E. (1927). "Gesammelte Abhandlungen." Springer-Verlag.
Hilbert, D. (1900). "Mathematische Probleme." Archiv der Mathematik und Physik, 1(1), 44-63.
Lang, S. (1993). "Real and Functional Analysis." Springer-Verlag.

Introduction

In our previous article, we explored the solution to Hilbert's 17th problem, known as Artin's solution, which states that any polynomial in a real field that is nonnegative at every point of the real closure of the field can be represented as a sum of squares of polynomials. In this article, we will answer some of the most frequently asked questions about Artin's solution and its implications.

Q: What is the real closure of a field?

A: The real closure of a field is a real field that contains the original field as a subfield and is such that every nonempty set of elements of the real closure that is bounded above has a least upper bound in the real closure.

Q: What is the significance of Artin's solution?

A: Artin's solution is significant because it provides a representation of nonnegative polynomials as sums of squares of polynomials, which has far-reaching implications for the study of real fields and their properties.

Q: What are some of the applications of Artin's solution?

A: Artin's solution has been used in a wide range of applications, including algebraic geometry and optimization theory. It has also been used to study the properties of nonnegative polynomials and to provide new insights into the study of real fields.

Q: Can you provide an example of a polynomial that can be represented as a sum of squares of polynomials?

A: Yes, consider the polynomial $f(x) = x^2 + 1$ . This polynomial is nonnegative at every point of the real closure of the field of real numbers, and it can be represented as a sum of squares of polynomials: $f(x) = (x + 1)^2$ .

Q: What are some of the limitations of Artin's solution?

A: Artin's solution is limited to polynomials that are nonnegative at every point of the real closure of the field. It does not provide a representation of polynomials that are nonnegative at some points but negative at others.

Q: Can you provide a proof of Artin's solution?

A: The proof of Artin's solution is based on a series of lemmas and theorems that establish the existence of a representation of nonnegative polynomials as sums of squares of polynomials. The key idea behind the proof is to show that if a polynomial $f$ is nonnegative at every point of the real closure $R$ of $E$ , then there exists a polynomial $g$ such that $f = g^2$ . This is done by using a series of algebraic manipulations and the properties of the real closure of a field.

Q: What are some of the open problems related to Artin's solution?

A: One of the open problems related to Artin's solution is to determine whether the representation of nonnegative polynomials as sums of squares of polynomials is unique. Another open problem is to determine whether Artin's solution can be extended to polynomials that are nonnegative at some points but negative at others.

Conclusion

In conclusion, Artin's solution to Hilbert's 17th problem is a significant result in the field of real fields and has far-reaching implications for the study of nonnegative polynomials. We hope that this Q&A article has provided a helpful overview of the solution and its implications.

References

Artin, E. (1927). "Gesammelte Abhandlungen." Springer-Verlag.
Hilbert, D. (1900). "Mathematische Probleme." Archiv der Mathematik und Physik, 1(1), 44-63.
Lang, S. (1993). "Real and Functional Analysis." Springer-Verlag.