DescriptionIn courses on Algebraic Number Theory and Galois Theory, the examples given in lectures and lecture notes can typically only scratch the surface as calculations become unwieldy very quickly. This entails that one merely gets a rather superficial understanding of the intricate notions and the amazing and beautiful relationships they enjoy. In light of some fantastic computer algebra software packages--for Number Theory I strongly recommend GP-PARI, a free, powerful and surprisingly intuitive package of routines that allow to distil information on the arithmetic of a given number field (units, ideals, class group, prime ideal decomposition, ...) and make them come to life'. What is more, the software allows to verify or illustrate key theorems and conjectures: Class number formula, Stark's Conjecture, Dirichlet's Unit Theorem, Class number formula, even the Galois action or class field theory as well as Kronecker's `Jugendtraum' (dream of youth) and much more can be made tangible.In this project, you are expected to build on your experience with Number Theory--and ideally also Galois Theory--and, after an initial `instigation period', to concoct your own examples of interesting number fields which allow to highlight as many of those topics as possible. I do not think that there is any introductory text for this out there. Instead you will have a huge choice and a wonderful playground at your disposal. Moreover, by comparing results for lots of cases of a similar `class' (say fixed number of real and of complex embeddings) you may be able to spot patterns that may not be clear from the outset or, if you are really lucky, which have not been discovered earlier. The project will almost invariably require that you learn to interact with some of the pertinent computer algebra packages that are available. The arguably most straightforward and convenient among them is GP/PARI (e.g. just type `bnfinit(x^2+53).clgp' to obtain, after 2ms, the answer to one of the recent exam questions). If you are used to working with Python then perhaps Sage is a good option--in fact, it is more powerful than GP/PARI but its number theoretic functionality is essentially GP/PARI itself. A more abstract software which unfortunately is not free and currently not supported by the university is Magma. But for short computations (that take less than a minute on a fast machine) there is an online `calculator' linked below. Magma takes some getting used to. PrerequisitesThis project is for students who are strongly interested in Number Theory and Galois Theory, and who also are willing to learn and [deal with] some computer algebra software (GP-Pari/Sage/Magma).
ResourcesLook for `Explicit (methods in) number theory' as well as `Computational number theory' for potential resources--note that the interpretation of those labels may differ widely.You can download GP/PARI from here and binaries for Sage (SageMath site) here. A Magma online calculator (only allows for calculations that do not exceed a minute's worth of time, possibly browser-dependent functionality) is here. Henri Cohen, the original creator of GP/PARI, wrote a book containing the background and algorithms for an earlier version of the package.
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email: Herbert Gangl