### On the existence of a non-principal Euclidean ideal class in biquadratic fields with class number two

##### Speaker: ** Mr. Sunil Kumar Pasupulati **, IISER Thiruvananthapuram

**Abstract**: In 1979, Lenstra introduced the definition of the Euclidean ideal which is a generalization of Euclidean domain.

**Definition 1.** Let R be a Dedekind domain and \mathbb{E} be the set of non zero integral ideals of R. If C is an ideal of R, then it is called Euclidean if there exists a function \psi : \mathbb{E} → \mathbb{N} , such that for every I \in \mathbb{E} and x \in I^{-1}C \ C there exist a y \in C such that

\psi ((x - y)IC^{-1})

H.Graves constructed an explicit biquadratic field

\mathbb{Q}(\sqrt{2}, \sqrt{35}) which has a non-principal Euclidean ideal class. C. Hsu generalized the result by Graves and proved suppose

K = \mathbb{Q}(\sqrt{q}, \sqrt{kr}) ,then

K has a non-principal Euclidean ideal class whenever

h_K = 2. Here the integers q, k, r are all primes ≥ 29 and are all congruent to 1 modulo 4. The family of biquadratic field given by C. Hsu is extended by Chattopadhyay and Muthukrishnan and proved that, If

K = \mathbb{Q}(\sqrt{q}, \sqrt{kr}), where

q ≡ 3 and

k, r ≡ 1 (mod 4) are prime numbers. Suppose that

h_K = 2. Then K has a Euclidean ideal class. In this talk, I will prove that a new family of biquadratic fields having non principal Euclidean ideal class whenever the class number of biquadratic field equal to two. This is joint work with Srilakshmi Krishnamoorthy.