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Advanced Number Theory

I'm eager to study some advanced topics in Number Theory, of special interest would be applying them to Olympiad problems. If anyone can recommend me some resources for learning and training, it will be a real pleasure. If I find something considered of interest, I also will share it.

What do I consider as advanced topics? First of all use of complex numbers, like Quadratic Extensions and other related topics, also of great interest would be usage of p-adic numbers, anything that will imply usage of abstract algebra, in particular Dirichlet L-functions and Finite Field Extensions.

If anyone got something interesting related to the topics listed above, please answer.

P.S. I'm not sure if Brilliant allows it, but feel free to contact me at nikolshapoval at gmail dot com. If it is against the rules, please let me know and I will remove this part.

Note by Nicolae Sapoval
4 years, 2 months ago

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First: how important is it to apply the advanced techniques to Olympiad-level problems?

If your answer is "very", you may be in for some disappointment since only very rarely do these techniques offer additional insight. A notable example I can recall is IMO 2008 Q3, Some more examples are given here but generally, such cases are scarce.

If you'd like to learn advanced topics on their own merit, then there's little choice but to follow the long arduous road of learning group theory, rings, fields, commutative rings, before embarking on algebraic number theory.

[ That being said, it's possible to learn unique factorization of "small" quadratic rings (e.g. ring of Gaussian integers) on a very rigourous basis without going into deep mathematics. ]

For p-adic numbers, you have to learn analysis and topology as well because the construction relies on a completely new idea of "distance" (technical term: metric) on the set of rational numbers.

Finally, I've some notes which may be relevant to what you're asking for, but not sure how to share my notes on Brilliant. Anyone knows? C Lim · 4 years, 2 months ago

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@C Lim Can you share it to me too? My email id is shivangjindal@live[dot]com Shivang Jindal · 4 years, 2 months ago

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@C Lim Thanks for your response. I already started diving into abstract algebra, of course I pursue the chance of better understanding modern mathematics. On other hand learning this matter now will save my time in undergrad and grad studies. I ask for Olympiad problems application both to see how this ideas may be applied on some contest problems and to give some reasoning for publishing articles (i.e. articles related to Olympiad techniques and their relationship with modern math).

It would be of great help if you would share your notes via Dropbox or e-mail/Google Drive. Thank you again for assistance. Nicolae Sapoval · 4 years, 2 months ago

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@Nicolae Sapoval I just emailed the notes to you.

But I really hope to upload them here so that more students can read them; also any mistakes can be pointed out. C Lim · 4 years, 2 months ago

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@C Lim You can send your notes to me, and I'd look at incorporating them onto the site. I started out developing some theory on Gaussian Integers, and will continue to build them out as I cover more basics. Calvin Lin Staff · 4 years, 2 months ago

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@Calvin Lin Thanks. I just emailed the notes to you (and also to all the others who requested in this thread).

Hope it's of use. C Lim · 4 years, 2 months ago

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@C Lim sir can you email it to me too?

email ID- vcpranav@yahoo.com Pranav Chakravarthy · 4 years, 2 months ago

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@C Lim Sir, can you please email it to me too? Email ID - mishraanoopam@yahoo.com Anoopam Mishra · 4 years, 2 months ago

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@C Lim Could you email the notes to me, my email is trivneel211@gmail.com Neelay Trivedi · 3 years, 8 months ago

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@C Lim Can you email them to me too, sir? Here's my email ID- akshatjain0205@yahoo.co.in Akshat Jain · 4 years, 2 months ago

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@C Lim sir can u please give me the notes id is d_bhattacharia@rediffmail.com Superman Son · 4 years, 2 months ago

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