I came across this theorem: For every ordinal $\alpha $ there exists a Euclidean ring whose Euclidean ordinal type is $\omega ^{\alpha }$. Moreover, these are the only possible Euclidean ordinal types for Euclidean rings
Each Euclidean ring has a unique minimal Euclidean norm which is "logarithmically superadditive" but not multiplicative (while the opposite is true for the absolute value). As a result, the naturalness of the superadditivity condition on the minimal norm is emphasized in the study of transfinitely valued norms. It has long been an open question whether every finitely evaluable Euclidean ring has some multiplicative Euclidean norm $\varphi :R\setminus 0 \to \mathbb{Z}_{\geq 0}$ such that $\varphi (xy)=\varphi (x)\varphi (y)$ for every pair $x,y\in R\setminus 0$
It is well known that any ordinal can be uniquely written in the normal Cantor form $$\omega ^{\alpha _1}+\omega ^{\alpha _1}n_2+\ldots+\omega ^{\alpha _k}n_k=\sum_{i=1}^{k}\omega ^{\alpha _i}n_i,$$ where $\alpha _1>\alpha _2>\ldots>\alpha _k$ are ordinals, the coefficients, $n_1$, $n_2\ldots, n_k$ are positive integers and $k\in \omega $
Can you write about which Cantor formula we are talking about? I couldn't find anything on the Internet about it
Note that your given $$\omega ^{\alpha _1}+\omega ^{\alpha _1}n_2+\ldots+\omega ^{\alpha _k}n_k=\sum_{i=1}^{k}\omega ^{\alpha _i}n_i,$$ where $\alpha _1>\alpha _2>\ldots>\alpha _k$ are ordinals, the coefficients, $n_1$, $n_2\ldots, n_k$ are positive integers and $k\in \omega $
(cit.) is incorrect for several reasons:
First, $\alpha _1>\alpha _2>\ldots>\alpha _k\geq0,$
second, is misses coefficient $n_1$, and the coefficients are non-zero natural numbers,
and last, $k$ is a natural, not an ordinal number. Perhaps this did throw you off?
Cantor published his results on ordinal arithmetic in two parts. He layed the foundation to his theorem in his 1895 paper Beiträge zur Begründung der transfiniten Mengenlehre, which can be found here, and in 1897 published the second part, which includes the notion of a normal form and its prove in $(§17)$, page 229, and can be found here.
He proved, although in different notation, that every ordinal number $\alpha\gt0$ can be uniquely written as a finite sum
$$\alpha=\omega ^{\beta _{1}}c_{1}+\omega ^{\beta _{2}}c_{2}+\cdots +\omega ^{\beta _{k}}c_k=\sum_{i=1}^{k}\omega ^{\beta _i}c_i $$
where $k$ is a natural number,
$c_{1},c_{2},\ldots ,c_{k}$ are non-zero natural numbers, and
$\beta _{1}>\beta _{2}>\ldots >\beta _{k}\geq 0$ are ordinal numbers.
This decomposition of an ordinal $\alpha$ is called the Cantor normal form of ordinal $\alpha$.
A variation which is usually easier to work with, is to set all the numbers $c_i$ equal to $1$ and subsequently allow the exponents to be equal. In other words $$\alpha=\omega ^{\beta _1}+\omega ^{\beta _2}+\ldots+\omega ^{\beta _k}=\sum_{i=1}^{k}\omega ^{\beta _i}$$
where $k$ is a natural number and
$\beta _{1}\geq \beta _{2}\geq \ldots \geq \beta _{k}\geq 0$ are ordinal numbers.
This is also called the Cantor normal form of an ordinal $\alpha$.
The beauty of this second definition is that the ordinal $\omega=\omega^\color{red}1$ and the natural number $\color{red}1=\omega^0$ have exactly $\color{red}1$ term in their Cantor normal form, and a finite number $n=\omega^0 +\ldots +\omega^0=\sum_{i=1}^{\color{red}n}\omega ^{0}$ has exactly $\color{red}n$ terms in its Cantor normal form.