In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.

Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.

However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.

This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.

However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.

This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

- Category
- number theory Mahematics
- Tags
- number theory, math

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