| Title |
The Explicit Isomorphism Problem on Global Fields / |
| Translation of Title |
Išreikštinio izomorfizmo problema globaliuose kūnuose. |
| Authors |
Montessinos, Mickael Gerard Bernard |
| DOI |
10.15388/vu.thesis.726 |
| Full Text |
|
| Pages |
177 |
| Keywords [eng] |
Explicit Isomorphism Problem ; Amitsur Cohomology ; Vector Bundles ; Algorithms |
| Abstract [eng] |
This dissertation studies the algorithmic problem of finding isomorphisms to matrix algebras over a global field. Over number fields, a polynomial quantum algorithm is provided under the Generalised Riemann Hypothesis. This relies on computing a presentation of a central simple algebra using Amitsur cohomology. Then, finding an isomorphism to a matrix algebra is equivalent to solving a certain equation. This equation is solved by computing S-units in an extension of the base field. Over function fields, existing results for algebras over a rational function field are given a new interpretation using the language of vector bundles. An algorithmic theory of vector bundles over algebraic curves as pairs of lattices is developed. This suggests an approach for solving the isomorphism problem over function fields of higher genera. |
| Dissertation Institution |
Vilniaus universitetas. |
| Type |
Doctoral thesis |
| Language |
English |
| Publication date |
2025 |
