Lucarno Adell, Pablo (2026) On Monogeneity of Elliptic Curves. Bachelor's Thesis, Mathematics.
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Abstract
This thesis studies the monogeneity of number fields attached to elliptic curves. While cyclotomic fields are always monogenic, much less is known about the division fields of elliptic curves. We develop the necessary background on monogeneity, elliptic curves, and division fields as Galois extensions of Q. To determine whether a number field is monogenic, we use Dedekind’s criterion. Hence, we analyse how primes split in these fields via the action of the Frobenius. In particular, by comparing the splitting behaviour of primes and the factorisation of polynomials in the finite fields, we identify primes that are common index divisors, which therefore obstruct monogeneity. We show that for elliptic curves without complex multiplication, the division fields Q(E[n]) are not monogenic for infinitely many n.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Ozman, E. and Kilicer, P. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 21 Jul 2026 12:39 |
| Last Modified: | 21 Jul 2026 12:39 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/37458 |
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