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I’ve at last finished (after an editing process interrupted by the Canford Summer School of Music for a week) the report for the Cambridge team at the International Maths Competition which took place last month in Budapest. It takes the traditional IMO leader format of a short serious report followed by a long amusing one. I hope you enjoy it.

[The report has been removed from online, but feel free to contact me if you want to read it].

Really just a note to remind me why the modules proof of JNF works.

Suppose is an algebraically closed field, and consider for some fixed automorphism of , a finite dimensional vector space over , the -module with scalar multiplication given by . It’s easy to check this yields the appropriate module structure.

Now, let’s think about what happens if . This is telling us that is nilpotent with degree . In this case, it is certainly true that there is a vector such that is nonzero, and furthermore, we must have that are linearly independent (if not then we can find some polynomial of degree less than which annihilates the entire module, which is clearly preposterous.

So let’s take these vectors. They are a basis, since it is clear the dimension of over is . But it is obvious that if we apply to an arbitrary vector we get

Writing this as a matrix,

.

This is a Jordan block. The JNF is now clear for an arbitrary linear map by writing (using the structure theorem for finitely generated modules) the above module as a direct sum of modules of the form above – utilising that is algebraically closed to note that all primes in are linear, and hence that the primary decomposition is of the desired form, splitting the space into Jordan blocks.