Algebraically Closed Fields Superseding the Complex Numbers

In mathematics, the concept of fields plays a crucial role in algebra and beyond. Among the most well-known fields is the set of complex numbers, denoted as ?. An algebraically closed field is one that has the special property of being closed under polynomial roots. This means that any non-constant polynomial with coefficients in that field has at least one root within the field itself. When considering fields that supersede the complex numbers, several interesting examples arise that meet this criterion.

Algebraic Closure of ?

The algebraic closure of ? is exactly ? itself, as ? is already an algebraically closed field. This means that all non-constant polynomials with coefficients in ? have a root in ?. This is a fundamental property of the complex numbers, established by Gauss's Fundamental Theorem of Algebra.

Field of Formal Power Series

A different example of an algebraically closed field that includes ? is the field of formal power series ?[[t]]. This field consists of infinite series of the form:

[sum_{n0}^{infty} a_n t^n, quad a_n in mathbb{C}]

The elements of this field are sums of complex numbers multiplied by powers of a variable t. Being a formal construction and not necessarily requiring a topology, ?[[t]] is an example of a field that is not just superseding but also extending ? in a significant way.

Field of Rational Functions

The field of rational functions ?(t) is another field that contains ?, but it is not algebraically closed in itself. However, by considering algebraic closures of its elements, ?(t) can be extended to an algebraically closed field. This extension involves adjoining roots of polynomials to the elements of ?(t), creating a larger field that includes all possible roots needed for closure.

Extensions of ?

Any algebraic extension of ? that is also algebraically closed will contain ? and can be made algebraically closed. For instance, the field of complex numbers adjoined with roots of all polynomials, like ?α for some algebraic α, can be constructed to ensure all necessary roots are included.

Consider the field extensions obtained by adjoining the roots of various polynomials to ?. These extensions are algebraically closed by definition, meaning they include all roots of polynomials with coefficients in the original field.

Transcendental Extensions

There are also transcendental extensions that can be algebraically closed. For example, the field ?(t1, t2, ...), the field of rational functions in infinitely many variables over ?, can be algebraically closed in certain contexts. This field expansion is significant as it allows for a broader range of algebraic structures while maintaining closure under roots of polynomials.

While these examples are rich in mathematical content, they demonstrate how the concept of algebraic closure can be extended beyond ? to encompass various algebraically closed fields.

The Conceptually Simplest Field Containing ?

From a different perspective, the conceptually simplest field that contains ? is the Field of Surcomplex numbers, No[i], where No is the field of surreal numbers. The surreal numbers are a rich and powerful construction within set theory that form a universal ordered field. Any ordered field can be embedded as a subfield of the surreal numbers, according to the principles of NBG set theory. Thus, the surcomplex numbers provide a vast and rich extension of the complex numbers.

Note, however, that while the surreal numbers are incredibly extensive, they are not as straightforward as algebraically closed fields. They come with additional structure and complexity related to their construction as a universal ordered field.

Conclusion

While the complex numbers ? are themselves an algebraically closed field, various extensions and related fields can serve as algebraically closed fields that include ?. These include fields such as the algebraic closure itself, fields of formal power series, fields of rational functions, certain algebraic extensions, and even constructions such as surreal numbers. Each of these fields offers unique insights into different areas of mathematics and how algebraic closure can be extended and understood.

Keywords: algebraically closed fields, complex numbers, surreal numbers, transcendental extensions