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Art of Mathematics, $2$nd Ed., p. 134, John Wiley & Sons, 1986,

Chapter 9: Fundamental Concepts, Parts 1-3, pp. 241-248,

Mathematics, A tool for science, E.B. Cheney and D.E. Newton, Chapter 11:
… the A = B theorem, the well-ordering theorem, and the proof of the fundamental theorem of arithmetic, pp. 248-249,

Mathematical Methods for Physicists, V1.1, Section 9.2, pp. 372,
and V1.2, Section 9.3, p. 383, Academic Press, 1980,

A:

Your first link already contains the answer. The three properties you mention are exactly what constitutes something being the most famous and therefore the best in a certain area.
Furthermore, it is worth to notice that e.g. Galois theory can be seen as being developed after the appearance of the fundamental theorem of arithmetic, because the latter was the basis of its proof.
Any field with the fundamental theorem of arithmetic is called an algebraically closed field.
However,

It is a common mistake to believe that all algebraically closed fields are known. However, the algebraically closed fields are merely a small subclass of the whole class of the algebraically closed fields.

Further, we can look for properties a field has to have if it is algebraically closed (so, in particular, if it is a field of characteristic zero), e.g. a commutative group of invertible elements $F^{*}$ that contains a multiplicative inverse of every element (i.e. every element of $F$ can be written as a product of invertible elements of $F$ with a commutative inverse) is required.
In fact, the result of this search is that we can use polynomials, more precisely polynomials over a field.
In his book The Art of Finite Fields, Garrett Birkhoff presents the problems of existence and uniqueness of a subfield in an algebraically closed field in addition to the existence of an irreducible polynomial.
The book demonstrates that the result is necessary, if one wants to define the

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SuanShu is an object-oriented numerical library. It has a high performance and exhaustively tested. This is the third edition of the first edition.
SuanShu 2.0 has been upgraded to use the latest third-generation numbers, the third edition of the new object-oriented numerical library. SuanShu is a numerical library mainly for numerical analysis and linear algebra. SuanShu’s classes are also being used to construct nonlinear, non-smooth, and nonlinear differential equations. This is a numerical library. Like a toy model, SuanShu is easy-to-use, but not easy to assemble.
The goals of this third edition are to present all the latest features, performance, and bug fixes. More importantly, as in the previous editions, SuanShu will always be aligned with the new development of traditional languages and their numerical technology.
SuanShu 2.0 is designed for Java environment. It is a pure C++ library. The core of SuanShu is a collection of classes. These classes can be connected in many ways to generate complex systems. This is a numerical library. Unlike Lego, you need little experience or mathematics knowledge to connect these classes. The amount of code you have to write is only the calling that uses these classes.
This is a very flexible numerical library. You can use these classes for either discrete mathematics, numerical analysis, or linear algebra. They are all included in this third edition, and they all have been updated.
SuanShu 2.0 is a Java library. It is based on the same concept as the NumPy 1.5. This Java version is tested thoroughly to ensure its compatibility with the C++ version. There are a lot of base classes of basic math operations, from which you can easily generate more complex things.
SuanShu comes with a demo version of basic classes, which you can quickly test out its usability and see its vast use.
SuanShu Demo:
To download the demo version of SuanShu, please click here.
Documentation:
To open the SuanShu 2.0 Documentation, please click here. The version of this documentation that comes with the demo version is updated.
Please contact the SuanShu 2.0 developers:
Development Team List:
Development Team Contact Name: Jingpeng Wu
Development Team Contact Email: jwu@suanshu.org
Development Team Contact Phone:
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Tutorials for SuanShu:

Linux tutorials for SuanShu:

Linux/WinNT/SunOS tutorials for SuanShu:

GNUPLOT users:
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JAVA Numerical Library SuanShu is a Java library of numerical methods and for numerical analysis.It is an object-oriented, high performance, extensively tested, and professionally documented library of numerical methods.
JAVA Numerical Library SuanShu contains a lot of algorithms found in Netlib and other public domains but they are refactored to become solidly object-oriented, unified and testable. SuanShu has for each mathematical concept, rather than procedure like our competitors’ products, a class for it. More importantly, these classes are designed so that they are very easy-to-use and easy-to-integrate.
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For him to assemble our classes to construct an application is as easy as a child assembles Lego pieces to construct a toy robot. Lego has simple pieces like bricks and gears and complex ones like mini-figures.
SuanShu Description:
JAVA Numerical Library SuanShu is a Java library of numerical methods and for numerical analysis.It is an object-oriented, high performance, extensively tested, and professionally documented library of numerical methods.
JAVA Numerical Library SuanShu contains a lot of algorithms found in Netlib and other public domains but they are refactored to become solidly object-oriented, unified and testable. SuanShu has for each mathematical concept, rather than procedure like our competitors’ products, a class for it. More importantly, these classes are designed so that they are very easy-to-use and easy-to-integrate.
They can be assembled and connected in many ways to construct more complicated concepts, hence new classes. The built-in and user-defined, old and new, simple and complex classes all conform to the same design standard to ensure that they are compatible.
SuanShu is to engineering solution as Lego is to toy model. SuanShu’s

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OS: Windows 7/8/8.1/10 (64-bit)
Processor: Intel Core i3 or later
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