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SUMMARY:Shabnam Akhtari (University of Oregon)
DTSTART:20211118T233000Z
DTEND:20211119T003000Z
DTSTAMP:20260422T053749Z
UID:SFUQNTAG/41
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SFUQNTAG/41/
 ">Orders in cubic and quartic number fields and classical Diophantine equa
 tions</a>\nby Shabnam Akhtari (University of Oregon) as part of SFU NT-AG 
 seminar\n\n\nAbstract\nAn order $\\mathcal{O}$ in an algebraic number fiel
 d is called monogenic if over $\\mathbb{Z}$ it can be generated by one ele
 ment. Győry has shown that there are finitely equivalence classes $\\alph
 a \\in \\mathcal{O}$ such that $\\mathcal{O} = \\mathbb{Z}[\\alpha]$\, whe
 re two algebraic integers $\\alpha\, \\alpha'$ are called equivalent if $\
 \alpha + \\alpha'$ or $\\alpha - \\alpha'$ is a rational integer. An inter
 esting problem is to count the number of monogenizations of a given monoge
 nic order. First we will note\, for a given order $\\mathcal{O}$\, that $$
 \\mathcal{O} = \\mathbb{Z}[\\alpha] \\text{ in } \\alpha$$ is indeed a Dio
 phantine equation. Then we will discuss how some old algorithmic results c
 an be used to obtain new and improved upper bounds for the number of monog
 enizations of a cubic or quartic order.\n\nThis talk should be accessible 
 to any math graduate student and\nquestions about basic concepts are welco
 me. We will start by recalling\nsome definitions from elementary algebraic
  number theory. Number\nfields\, lattices over $\\mathbb{Z}$\, and simple 
 polynomial equations are the main\nfocus of this talk.\n
LOCATION:https://researchseminars.org/talk/SFUQNTAG/41/
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