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$\color{Green}{Background:}$

The Division Algorithm was a key tool in analyzing the arithmetic of both $\mathbb{Z}$ and $F[x].$ So we now look at domains that have some kind of analogue of the Division Algorithm. To see how to describe such an analogue, note that the degree of a polynomial in $F[x]$ can be thought of as defining a function from the nonzero polynomials in $F[x]$ to the nonnegative integers. By identifying the key properties of this function, we obtain this

$\textbf{Definition:}$ An integral domain $R$ is a Euclidean domain if there is a function $\delta$ from the nonzero elements of $R$ to the nonnegative integers with these properties;

$(i)$ If $a$ and $b$ are nonzero elements of $R,$ then $\delta(a)\leq \delta(ab).$

$(ii)$ If $a,b\in R$ and $b\neq 0_R,$ then there exist $q,r\in R$ such that $a=bq+r$ and either $r=0_R$ or $\delta(r) < \delta(b).$

$\color{Red}{Questions:}$

From the above motivation for Euclidean domain, I want to know what happens if we try to have an analogue of the division algorithm for an integral domain without the use of the $\delta$ function. I try to look this up in various algebra text and also online but was not able to find a satisfactory answers. It feels like the use of the $\delta$ function is some extra gadget that will allow for division in an integral domain. I am sorry in advance if I am not using the correct phrasing in asking my question.

Thank you in advance

Moo
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Seth
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  • Well, there's the whole Grobner basis thing that applies to polynomials in several variables with coefficients in a field. Those are inherently non-Euclidean domains. There you have "monomial orders" and guaranteeing uniqueness requires some work. An introduction to the subject is in Ideals, Varieties and Algorithms by Cox, Little, O'Shea. – Sassatelli Giulio Dec 18 '23 at 21:13
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    You do need that "gadget" or something like it to measure the "size" of ring elements. That's because the Euclidean algorithm is essentially an induction on the size. This may count as an answer: https://math.stackexchange.com/questions/3379695/why-does-the-euclidean-algorithm-for-finding-gcd-work/3379763#3379763 – Ethan Bolker Dec 18 '23 at 21:18
  • @EthanBolker can you elaborate more on what you mean by the $\delta$ function for measuring the "size" of ring elements? – Seth Dec 19 '23 at 02:42
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    @Seth Your points i and ii precisely describe with inequalities just what you need to make the argument in my linked answer work. Integers a big when their absolute value s big. Polynomials are big when they have high degree. In algebraic number theory numbers are big when their norms are big. – Ethan Bolker Dec 19 '23 at 12:53
  • @EthanBolker but why not just use the norm function as the $\delta$ function instead? Or is using the $\delta$ function allows for more different kind of function that have the i and ii properties? – Seth Dec 19 '23 at 16:41
  • You do use the norm function when it works. It always satisfies i. When it satisfies ii you have unique factorization. See https://en.wikipedia.org/wiki/Euclidean_domain#Norm-Euclidean_fields – Ethan Bolker Dec 19 '23 at 18:32

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