Chennai Mathematical Institute


3.30 pm, Seminar Hall
Euclidean algorithm for certain algebraic number fields

Subramani Muthukrishnan
Chennai Mathematical Institute.


Let $K$ be an algebraic number field with the ring of integers $\mathcal{O}_K$. We say that $K$ is Euclidean if there exists a Euclidean function $\phi : \mathcal{O}_K \to \mathbb{N} \cup \{0\}$ defined as follows: $\phi (\alpha) = 0 \textrm{ if and only if } \alpha = 0$, and for all $\alpha, \beta \not = 0 \in \mathcal{O}_K$ there exists a $\gamma \in \mathcal{O}_K$ such that $\phi(\alpha - \beta \gamma) < \phi(\beta).$

One of the classical problem in algebraic number theory is to classify number fields which are Euclidean. For quadratic fields the answer is completely known when $\phi$ is the \emph{norm} function. However, when $K$ is real quadratic field and $\phi$ is not the norm-function, then very little is known.

In this talk, we shall discuss the following results: 1) We shall exhibit an infinite family of real quadratic fields in which a field is Euclidean if and only if it has class number one, 2) To show that any real quadratic field is Euclidean provided Hardy-Littlewood conjecture and Wieferich prime conjectures holds in that field, 3) We shall also discuss the Euclidean problem for cyclic cubic fields. If time permits, we shall present an argument to show that there exists infinitely many non-Wieferich primes in $K,$ under \emph{abc} conjecture for number fields.