The Minimal Euclidean Function on the Gaussian Integers

Motzkin proved that every Euclidean domain R has a minimal Euclidean function, $\phi_R$ . He showed that when $R = \mathbb{Z}$, the minimal function is $\phi_{\mathbb{Z}} (x) =\ log 2_ |x|$.  For over seventy years, $\phi_{\mathbb{Z}}$ was the only example of an explicitly-computed minimal function in a number field. We give the first explicitly-computed minimal function in a non-trivial number field, $\phi_{\mathbb{Z}[i]}$. The proof introduces a new way to visualize quotients $\mathbb{Z}[i]$.  We also present the first division algorithm for $\Z[i]$ relative to $\phi_{\Z[i]}$, empowering audience members to perform the Euclidean algorithm on the ring using its minimal Euclidean function.  This talk is accessible to undergraduates.

Hester Graves received her MA and Ph.D. in mathematics from the University of Michigan. She was a John Coleman Postdoctoral Fellow at Queen’s University and is a Research Staff Member at the Center for Computing Sciences.