### Session S09 - Number Theory in the Americas

Wednesday, July 14, 12:30 ~ 13:00 UTC-3

## When is the ring of integers of a number field coverable?

### Omar Kihel

#### Brock U., Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.

It is easy to see that a group cannot be the union of two of its proper subgroups. Scorza showed that a group is a union of three of its proper subgroups if and only if it has a quotient isomorphic to the Klein 4-group $V=C_2^2$. Similar results exist for coverings by four, five, and six proper subgroups. Consideration of a covering by seven proper subgroups yields a result akin to the two proper subgroups case: no group can be written as a union of seven of its proper subgroups. Few authors have considered the problem of covering a ring by its proper subrings. We say that a ring $R$ is coverable if $R$ is equal to a union of its proper subrings. If this can be done using a finite number of proper subrings, then $\sigma(R)$ denotes the \emph{covering number} of $R$, which is the minimum number of subrings required to cover $R$. We set $\sigma(R)=0$ if $R$ is not coverable, and we set $\sigma(R)=\infty$ if $R$ is coverable but not by a finite number of proper subrings. In this talk, among other results, we will answer to the two following questions:\\ \indent (1) When is the ring R of algebraic integers of a number field finitely coverable?\\ \indent (2) Calculate $\sigma(R)$?