Description

Mathematics Colloquium

Date and Time: Wednesday, 18 January, 2023, 4 pm

Host: Sudhir Ghorpade

Speaker: Sudesh Kaur Khanduja

Affiliation: Panjab University, Chandigarh

Title: When is Z[θ] the ring of integers?

Abstract: Let K = Q(θ) be an algebraic number field with θ an algebraic integer having minimal polynomial f(x) over Q. Let AK denote the ring of algebraic integers of K. In this talk, we shall discuss some necessary and sufficient conditions to be satisfied by f(x) so that AK = Z[θ]. In particular when f(x) is an irreducible trinomial x^n+ax^m +b ∈ Z[x], then we shall describe a set of necessary and sufficient conditions in terms of prime powers dividing a, b, m and n, for any prime p to divide the group index [A_K : Z[θ]]. Using the well known Dedekind Criterion, we shall also discuss a generalisation of this result for a simple ring extension R[η] of a valuation ring R to be integrally closed when η is a root of an irreducible trinomial x^n+ax^m +b belonging to R[x]. The latter result yields interesting number theoretic applications. This is partly based on joint works with A. Jakhar, B. Jhorar, Sumandeep Kaur, M. Kumar, and N. Sangwan.

Description

Ramanujan Hall, Department of Mathematics

Date

Wed, January 18, 2023

Start Time

4:00pm IST

Priority

5-Medium

Access

Public

Created by

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Updated

Fri, January 13, 2023 1:43pm IST